water reactor safety- research--analysis development (nrg

Distr ibution Category: Water Reactor Safety-

Resea rch - -Ana ly s i s Development (NRG-4)

ANL-77-47

ARGONNE NATIONAL LABORATORY 9700 South Cass Avenue Argonne, Ill inois 60439

ONE-DIMENSIONAL DRIFT-FLUX MODEL AND CONSTITUTIVE EQUATIONS FOR

RELATIVE MOTION BETWEEN PHASES

IN VARIOUS TWO-PHASE FLOW REGIMES

by

M. Ishii

Reactor Analysis and Safety Division

October 1977

NOTICE

This report was prepared as an account of work sponsored by the United States Government Neither the United States nor the United States Department of Energy, nor any of thetr employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibdity for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe pnvately owned nghts

BISTRIBUTION OF THIS DOCUMENT IS UNUMITED

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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TABLE OF CONTENTS

Page

NOMENCLATURE 6

ABSTRACT 9

I. INTRODUCTION 9

II. THREE-DIMENSIONAL FORMULATION 10

III. MULTIPARTICLE SYSTEM IN INFINITE MEDIA 12

IV. LOCAL DRIFT VELOCITY OF DISPERSED FLOW IN

CONFINED CHANNEL 20

V. ONE-DIMENSIONAL DRIFT-FLUX MODEL 22

VI. ONE-DIMENSIONAL DRIFT VELOCITY 25

A. Dispersed Two-phase Flow 25 B. Annular Two-phase Flow 38

C. Annular Mist Flow 42

VII. COVARIANCE OF CONVECTIVE FLUX 45

VIII. FLOW-REGIME TRANSITION AND DRIFT VELOCITY IN

VERTICAL SYSTEM 50

IX. CONCLUSIONS 53

APPENDIX: Relative Motion in Single-par t ic le System 55

ACKNOWLEDGMENTS 58

REFERENCES 58

LIST OF FIGURES

No. Title Page

1. Single-par t ic le Drag Coefficient 14

2. Termina l Velocity 14

3. Mixture-v iscos i ty Model 16

4. Comparison of P red ic ted Resul ts with Richardson and Zaki ' s Empi r i ca l Corre la t ions for Sol id-part iculate Flow Systems 17

5. Comparison of P r e s e n t Model with Exper imenta l Data for

Solid P a r t i c l e s 17

6. Exper imenta l Data for H igh -p re s su re System 27

7. Freon-22 Data in Boiling Flow 27

8. Fully Developed Nz-NaK Flow 28

9. Fully Developed Ai r -Wate r Flow Data 28

10. Fully Developed Nz-Mercury Flow Data 28 11. Exper imenta l Data for Cocurrent Upflow and Cocurrent Down-

flow of S team-Water System 28

12. Exper imenta l Data for Cocurrent Upflow and Cocurrent Down-flow of Heated Santowax-R System 29

13. Limiting Value of Distribution P a r a m e t e r CQ at Zero Void Frac t ion and Pg/pf -* 0 Based on Single-phase Turbulent-flow Profi le 30

14. Distribution P a r a m e t e r for Fully Developed Flow in a Round Tube 31

15. Distribution P a r a m e t e r for Fully Developed Flow in a Rectangular Channel 31

16. Axial Void Distribution with Corresponding Void Prof i les at Various Stations for S team-Water Exper iment in Rectangular Channel 32

17. Axial Void Distribution with Corresponding Void Prof i les at Various Stations for Freon-22 Exper iment in Round Tube 32

18. Exper imenta l Data of Rouhani and Becker in Round Tube and Effect of Developing Flow due to Boiling 33

19. Exper imenta l Data of St. P i e r r e in Rectangular Flow and Effect of Developing Flow due to Boiling 33

20. Exper imenta l Data of M a r c h a t e r r e in Rectangular Duct and Effect of Developing Flow due to Boiling 33

LIST OF FIGURES

No. Title Page

21. Distribution P a r a m e t e r in Developing Flow due to Boiling 34

22. Comparison of Annular-flow Corre la t ion with Exper imenta l Data . 41

23. Comparison of Data for Onset of Ent ra inment according to Inception Cr i t e r i a 52

LIST OF TABLES

No. Title Page

I. Summary of Drift Velocity V^j in Infinite Media 19

II. Drift Velocity for Ver t ica l Boiling System 53

6

N O M E N C L A T U R E

A To ta l flow a r e a h-^

A(j P r o j e c t e d a r e a of a p a r t i c l e h ^ s

a P a r a m e t e r def ined by Eq. 89 j c o r e

b P a r a m e t e r def ined by Eq. 89 ^ j f ) t r

C D D r a g coef f ic ien t for m u l t i p a r t i c l e s y s t e m j j

CDCO D r a g coef f ic ien t for s i n g l e - p a r t i c l e s y s t e m JQ

Co D i s t r i b u t i o n p a r a m e t e r j

Coo A s y m p t o t i c va lue of CQ jj^.

Chk E n t h a l p y - d i s t r i b u t i o n p a r a m e t e r for k p h a s e K

C-hm E n t h a l p y - d i s t r i b u t i o n p a r a m e t e r for m i x t u r e m

C-yk M o m e n t u r r k d i s t r i b u t i o n p a r a m e t e r fo r k p h a s e ^i]f.

Cvm M o m e n t u m - d i s t r i b u t i o n p a r a m e t e r for m i x t u r e M r k

c P a r a m e t e r def ined by Eq. 89 M r m

Cj3 D r a g coef f ic ien t in c h u r n - f l o w r e g i m e b a s e d M*

on Vdj

Ct|tk D i s t r i b u t i o n p a r a m e t e r for p r o p e r t y i|rk

D H y d r a u l i c d i a m e t e r E(j A r e a f r a c t i o n of l iqu id e n t r a i n e d in the g a s c o r e NRgoo

b a s e d on l iqu id a r e a

f(Q'(j) Given funct ion of void f r a c t i o n

f j^ M i x t u r e f r i c t i on f a c t o r

fi I n t e r f a c i a l f r i c t i o n f a c t o r

^wf Wal l f r i c t i o n f a c t o r for l iqu id f i lm

F n P r e s s u r e f o r c e

F g G r a v i t y f o r c e

Fj3 D r a g f o r c e

G To ta l m a s s flow r a t e

g G r a v i t y

g^ Ax ia l c o m p o n e n t of g r a v i t y

g* Nondimensioiiu.1 g r a v i t y p a r a m e t e r

h]^ E n t h a l p y of k p h a s °

h m E n t h a l p y of m i x t u r e

h m s a t E n t h a l p y of s a t u r a t i o n m i x t u r e

ho En tha lpy a t c e n t e r l i n e

N^

Np,f

NRe

n

P

P i

Pwf

qw

. ?.,.».. qT

qcr

Qk

R

Rw

Ref

Enthalpy at wall

Saturation enthalpy of k phase

Total volumetric flux based on the core a rea

Liquid flux at the laminar turbulent transit ion

Nondimensional gas flux; jg/vApgD/p^

Volumetric flux at centerline

Mixture volumetric flux

Volumetric flux of k phase

Kutateladze number

Exponent of power profile of propert ies

Interfacial drag force of k phase

Transverse s t r e s s gradient of k phase

Mixture t r ansver se s t r ess gradient

Nondimensional s t ress -grad ien t parameter

Viscosity number based on continuous-phase viscosity

Viscosity number based on liquid viscosity

Single-particle Reynolds number

Par t ic le Reynolds number

Exponent of power profile of proper t ies

P r e s s u r e

Wetted pe r imete r of interface

Wetted per imeter of wall for liquid phase

Wall heat flux

Conduction heat flux

Turbulent heat flux

feitical heat flux

Volumetric flow of k phase

Radial coordinate

Par t ic le radiJis

Nondimensional radius

Radius of pipe

F i lm Reynolds number

NOMENCLATURE I

N O M E N C L A T U R E

t T i m e

vfc L iqu id d r o p v e l o c i t y m a n n u l a r c o r e

vff L iqu id f i lm v e l o c i t y

Vgc G a s - c o r e v e l o c i t y

V(jj L o c a l d r i f t v e l o c i t y

((V(jj)) Weigh ted m e a n of l o c a l d r i f t v e l o c i t y

Vdj M e a n t r a n s p o r t d r i f t v e l o c i t y

Vj^ M i x t u r e v e l o c i t y

vk V e l o c i t y of k p h a s e

Vj. R e l a t i v e v e l o c i t y

Vj-a, T e r m i n a l v e l o c i t y for s i n g l e - p a r t i c l e sys tenn

V* N o n d i m e n s i o n a l v e l o c i t y

V(j V o l u m e of p a r t i c l e

y D i s t a n c e f r o m the w a l l

z Ax ia l c o o r d i n a t e

a Gas (vapor ) vo id f r a c t i o n

ay^ Void f r a c t i o n of k p h a s e

•^dm M a x i m u m pack ing void f r a c t i o n

< dw Void f r a c t i o n of d i s p e r s e d p h a s e a t w a l l

• do Void f r a c t i o n of d i s p e r s e d p h a s e a t c e n t e r l i n e

"^core F r a c t i o n of a r e a o c c u p i e d by a n n u l a r c o r e

•^drop F r a c t i o n of a r e a o c c u p i e d by d r o p l e t s f r o m a n n u l a r c o r e a r e a

(%^ Void f r a c t i o n of k p h a s e a t wa l l

cq -o Void f r a c t i o n of k p h a s e a t c e n t e r l i n e

T]^ M a s s S o u r c e for k p h a s e

6 T h i c k n e s s of l iqu id f i lm

Acf}^ D i f f e r e n c e of void f r a c t i o n a t c e n t e r and wa l l

Atdc P r o p e r t y d i f f e r ence b e t w e e n d and c p h a s e s

Ah^c E n t h a l p y d i f f e r e n c e , hd - he

Ap A b s o l u t e v a l u e of d e n s i t y d i f f e r e n c e

(ij V i s c o s i t y of k p h a s e

Mrn M i x t u r e v i s c o s i t y m d i s p e r s e d flow

?h

I

Pk

Pm

a

T

fT

^k

=J ^ki

T i

Twf

T T zz fM' m

*

*k

*m

H e a t e d p e r i m e t e r

R a t i o of w e t t e d p e r i m e t e r s , P i / P w f

D e n s i t y of k p h a s e

D e n s i t y of m i x t u r e

S u r f a c e t e n s i o n

A v e r a g e d v i s c o u s s t r e s s for m i x t u r e

T u r b u l e n t s t r e s s for m i x t u r e

A v e r a g e d v i s c o u s s t r e s s for k p h a s e

T u r b u l e n t s t r e s s for k p h a s e

A v e r a g e i n t e r f a c i a l s t r e s s for k p h a s

A x i a l s h e a r f o r c e a t a n n u l a r i n t e r f a c e

Wal l s h e a r f o r c e for l iqu id f i lm

N o r m a l v i s c o u s s t r e s s

N o r m a l t u r b u l e n t s t r e s s

M i x t u r e - e n e r g y d i s s i p a t i o n

Given funct ion of r j

P r o p e r t y of k p h a s e

M i x t u r e P r o p e r t y

S u b s c r i p t s

c Con t inuous p h a s e

d D i s p e r s e d p h a s e

f L iqu id p h a s e

g V a p o r o r g a s p h a s e

k k p h a s e (k = c,d o r k =

ki k p h a s e a t i n t e r f a c e

m M i x t u r e

S y m b o l s

= g.f)

(\li) A r e a a v e r a g e

((i|rj5-)) Weigh ted m e a n v a l u e

• m Weigh ted m e a n m i x t u r e p r o p e r t y

COV C o v a r i a n c e

* N o n d i m e n s i o n a l

ONE-DIMENSIONAL DRIFT-FLUX MODEL AND CONSTITUTIVE EQUATIONS FOR

RELATIVE MOTION BETWEEN PHASES IN VARIOUS TWO-PHASE FLOW REGIMES

by

M. Ishii

ABSTRACT

In view of the prac t ica l importance of the drift-flux model for two-phase flow analysis in general and in the analysis of nuc l ea r - r eac to r t rans ien ts and accidents in par t i cu la r , the kinematic constitutive equation for the drift velocity has been studied for var ious two-phase flow reg imes . The const i tutive equation that specifies the re la t ive motion between phases in the drift-flux model has been derived by taking into account the interfacial geometry, the body-force field, shear s t r e s s e s , and the interfacial momentum t rans fe r , since these m a c r o scopic effects govern the relat ive velocity between phases . A compar ison of the model with var ious exper imental data over var ious flow reg imes and a wide range of flow p a r a m e t e r s shows a sat isfactory agreement .

I. INTRODUCTION

Two-phase flows always involves some rela t ive motion of one phase with respec t to the other; there fore , a two-phase-flow problem should be formulated in t e r m s of two velocity fields. A genera l t rans ien t two-phase -flow problem can be formulated by using a two-fluid model or a drift-flux model, depending on the degree of the dynamic coupling between the phases . In the two-fluid model , each phase is considered separa te ly ; hence the model is formulated in t e r m s of two sets of conservat ion equations governing the balance of m a s s , momentum, and energy of each phase . However, an in t ro duction of two momentum equations in a formulation, as in the case of the two-fluid model , p resen t s considerable difficulties because of mathemat ica l complications and of uncer ta in t ies in specifying interfacial in teract ion t e r m s between two phases . " Numer ica l instabil i t ies caused by improper choice of in ter fac ia l - in terac t ion t e r m s in the phase -momentum equations a r e common; therefore careful studies on the interfacial constitutive equations a r e required in the formulation of the two-fluid model . For example, it has been suggested^ that the interact ion t e r m s should include f i r s t - o rde r t ime and spatial de r iva t ives under cer ta in conditions.

These difficulties associa ted with a two-fluid model can be significantly reduced by formulating two-phase problems in t e r m s of the drift-flux model, in which the motion of the whole mixture is expressed by the m i x t u r e -momentum equation and the re la t ive motion between phases is taken into account by a kinematic constitutive equation. Therefore , the basic concept of the drift-flux model is to consider the mixture as a whole, r a ther than as two separa ted phases . The formulation of the drift-flux model based on the mix tu re balance equations is s imple r than the two-fluid model based on the separa te balance equations for each phase. The most important assumption associated with the drift-flux model is that the dynamics of two phases can be expressed by the mix tu re -momentum equation with the kinematic constitutive equation specifying the re la t ive motion between phases . The use of the drif t-flux model is appropr ia te when the motions of two phases a r e strongly coupled.

In the drift-flux model , the velocity fields a r e expressed in t e r m s of the mixture cen t e r -o f -mass velocity and the drift velocity of the vapor phase, which is the vapor velocity with respec t to the volume center of the mix ture . The effects of t he rma l nonequilibrium a re accommodated in the drift-flux model by a constitutive equation for phase change that specifies the ra te of m a s s t r ans fe r per unit volume. Since the ra tes of m a s s and momentum t r a n s fer at the interfaces depend on the s t ruc tu re of two-phase flows, these const i tutive equations for the drift velocity and the vapor generation a r e functions of flow regimes.^ '*

The drift-flux model is an approximate formulation in compar ison with the m o r e r igorous two-fluid formulation. However, because of its s i m plicity and applicability to a wide range of two-phase-flow problems of p r a c t ical in te res t , the drift-flux model is of considerable impor tance . In par t i cu la r the model is useful for t rans ien t thermohydraul ic and accident analyses of both LWR's and LMFBR's , In view of the prac t ica l importance of the drift-flux model for two-phase-flow analyses , the kinematic constitutive equation for the drift velocity has been studied in the present analys is . The constitutive equation for the vapor-drif t velocity for bubbly and slug flows has been studied by Zuber et al. ~ by balancing the gravity force with the drag force. The drift velocity in two-phase annular flows has been analyzed.'^ The present study is an extension of the above analys is .

II. THREE-DIMENSIONAL FORMULATION

The three-d imens iona l form of the drift-flux model* has been obtained by using the t ime or s ta t is t ica l averaging method. The resul t can be s u m m a rized as follows:

Mixture Continuity Equation:

SPm 3t + V-(pmVm) = 0. (1)

Continuity Equation for Dispersed Phase :

^Q'dPd 3t + V-(adPdVm)

^dPdPc-* (2)

Mixture Momentum Equation:

St + '7-(PmVmVm) = "^Pm + V' U + = T 'd PdPc

1 - °'d P m ^dj^dj j + Pmg-

(3)

Mixture Enthalpy-energy Equation:

ap™h m " m 9t

+ V-(prnhrnVm) = -V-.T , "'dPdPc

Pm q + q' ^ + —:: (hd - hc)V(jj m

St

V m + ^d(Pc - Pd^

V ' m

dj • Pm + *S m ' 'm-(4)

Here aj^, pj^, hj^, T, and q a r e the conventional t i m e - (or ensemble) -averaged local void fraction, phase density, phase enthalpy, viscous s t r e s s , and conduction heat flux, respect ive ly . The component k denotes e i ther the continuous (k = c) or the d i spersed phase (k = d); Pm, VJYI, Pm> and hm a r e the local mean density, velocity, p r e s s u r e , and enthalpy of the mix ture , respect ively; f-* and q"-'- a r e the turbulent diffusion flux of momentum and energy for the mixture ; r, is the m a s s - t r a n s f e r t e r m for phase k due to phase changes; and ^ ^ ^ the energy-diss ipat ion t e r m . The velocity V jj denotes the drift velocity of the d i spersed phase; its physical significance is explained below.

The cen t e r -o f -mass velocity of the mixture is given by

vm = [o^dPd^d + (1 - »d)pcVc]/Pm. (5)

where v j and v^ a r e the mass-weighted mean velocit ies of phases d and c. The velocity of the center of volume of the mix tu re is

J - Q'dvd + (1 - ad)vc = Jd + J c (6)

where j ^ and Jc a r e the local volumetr ic fluxes of the indicated phases . The vapor drift velocity of a d ispersed phase is the velocity of the d i spersed phase with respec t to the volume center of the mixture :

Vd J Vd - r = (1 - Q'd)(vd - Vc) = (1 - Q'd)vr, (7)

where Vj. is the re la t ive velocity between phases . In Eqs . 2-4, the convective t e r m s a r e formulated in t e r m s of v ; the re fore , an additional diffusion flux due to relat ive motion between phases appears on the r ight-hand side of the equations. To take these diffusion effects into account, the drift velocity should be specified by kinematic constitutive equations, which may be wri t ten as

"^dj " '^dj^'^d' Pm' g' v ^ ' etc.) . (8)

To take into account the m a s s t rans fe r ac ros s the in terfaces , a constitutive equation for F^ should also be given. In a functional form, this phase-change constitutive equation naay be wri t ten as

d ~ 'd^^d'i^m' "m' ^^ ^ ^ - ^I^^'d'Pm' ^m' " i r ' V - V s a t ' «tc.). (9)

The vapor-dr i f t velocity in the drift-flux model plays a role s imi la r to that of the diffusion coefficient in a s ingle-phase two-component sys tem. However, the application of a diffusion coefficient is useful only when the r e l a tive motion between components or phases is due to a concentration gradient and can be expressed by a l inear constitutive law. For general two-phase flow sys tems , F ick ' s law of diffusion may not hold, since in this case the interfacial geometry, the body-force field, and the interfacial momentum t rans fe r a r e the factors governing the relat ive motion of phases . In other words , in two-phase sys tems , the diffusion of phases is macroscopic , whereas in s ingle-phase two-component sys t ems , it is due to the microscopic molecular diffusion. The constitutive equation for the vapor-drif t velocity is therefore expected to de pend strongly on the two-phase flow reg imes , since the momentum t rans fe r between the phases is governed by the geometry of the interfaces as well as by the in te r fac ia l -a rea concentration.

III. MULTIPARTICLE SYSTEM IN INFINITE MEDIA

The relat ive motion between phases can be studied by considering the momentum equations for each phase in the two-fluid-model formulation. In a th ree-d imens iona l form, the k-phase momentum equation'* is given by

^kPkV-57 ^ ^k'Vvkj = -^k^Pk + Q k -V k + "k ) + Q'kPkg + (Pki - Pk) Q'k

- ( -k - ki - ( k + ^k)] + ( ki - \) k + ^ik-(10)

where the subscr ipt ki denotes the value at the interface for phase k and Mj j is the interfacial drag force. Under the assumption that the averaged p r e s s u r e and s t r e s s in the bulk fluid and at the interface a r e approximately the s ame , we obtain

"kPki-aT * ~k'^~kj ' -»k'Pk + »k''(fk + ' k ) + "kPkB + " ik

+ < V - ' k f r k - <" '

The conservat ion of the mixture momentum requi res

ZMik = 0, (12) k

which is the modified form of the averaged momentum-jump condition.

The two-fluid formulation for d ispersed two-phase flow sys tem was used by C a r r i e r and Rannie, among o thers , in relat ion to low-concentrat ion flows through rocket nozzles . The problems were analyzed by considering the equations of motion for a par t ic le and mix ture with a relat ively s imple interact ion t e r m between phases . However, for flows with high concentrat ions and under rapid t rans ien t conditions, some modification of the analysis is required to apply the two-fluid model.

Now let us consider the derivat ion of the drift velocity from the two-fluid model. In the absence of the wall and under a s teady-s ta te condition without phase change (Pk = O), the mul t ipar t ic le sys tem in an infinite medium essent ia l ly reduces to a gravi ty-dominated one-dimensional flow, since the average void and velocity profiles become flat. Then the axial component of the momentum equation for k phase can be wri t ten as

° ^ " ° ' k - i f -^kPkg + ik- (13)

Here we have also assumed that the sur face- tens ion effect can be neglected, and therefore Pc ~ Pd ~ Pm* ^ ^ adding the phase -momentum equations and using Eq. 12, we obtain

ig. (14) dz •me

The drag force acting on the par t ic le under s teady-s ta te condition can be given in t e r m s of the drag coefficient Cy, based on the re la t ive velocity as

^D = -iC^PcVrKlA^, (15)

where A j is the projected a r ea of a typical par t ic le . Then Fj-j is re lated to the interfacial force by

F D = M. ,Vd/Q'd. (16)

where V, is the volume of a par t ic le . Then from Eqs . 13-15, we obtain

I . 8 ^d Vr |Vr (Pc - Pd)g(l - ^'d)' 3 C]3P^

where the mean radius of the par t ic le is defined by rd = 3V(j/(4A(j).

(17)

On the other hand, for a s ing le-par t ic le sys tem in an infinite medium, the force balance (as shown in the appendix) reduces to

" j ; °o I roo ' 8 ""d , .

^ ^DooPc (18)

Where v and C-Q^ a r e the t e rmina l velocity and drag coefficient of a single par t ic le in an infinite medium. In general , the drag law for a single par t ic le can be expressed by CDOO = CDco(Np^e<»)' where the Reynolds number NRgoo is given by NRe<» - 2rdPc|vr»|/Mic- The express ions for CDOO in var ious flow reg imes a r e given in the appendix and shown in Fig. 1. Also, the nondimensional t e rmina l velocity is shown as a function of the dimensionless radius of a par t ic le in Fig. 2.

100 r r r T T T i — i i i i i i i i i — i i i i i i i i i — i i i i i i i i i — i i i i i i w

(24/N„,„)f.O.IN°;^]

DROPLET LIMIT CAP BUBBLE

SOLID PARTICLE •

I I Mill i _ L i IIIIII I I I I m i l I I I I m i l I I I I I I

<

T i T i — I I I I I I I I I — I I I m i l l I I I m i l l ,

SOLID PARTICLE-

BUBBLE-

i m j I iiJ I I I I I I I m i l l I I I I I I

F i g .

SINGLE PARTICLE REYNOLDS NUMBER, Npj,„

1. Single-particle Drag Coefficient. ANL Neg. No. 900-76-443. Corr.

Then from Eqs . 17 and 18, we get

2 /

10' 10' 10 REDUCED RADIUS, r'= r ( p ^ g i / ) / ^ ^ )

Fig. 2. Terminal Velocity. ANL Neg. No. 900-76-445.

CD»(NReco) = CD(NRg) I-oo

(1 -«rf), (19)

where the Reynolds number is given by Nj^g = 2rdPcrr|/M'm - d lirn is the mix ture viscosi ty. If we know the mixture viscosi ty and the dependence of Cj) on Nj^g, Eq. 19 can be solved for the slip velocity in t e r m s of the s ingle-par t ic le t e rmina l velocity.

The p r e s e n c e of the m i x t u r e v i s c o s i t y in the s i m i l a r i t y g roup N R ^ I S exp la ined as follows:' ' '^^ A s ing l e p a r t i c l e mov ing t h r o u g h a d i s p e r s e d t w o -p h a s e m i x t u r e i m p a r t s a m o t i o n to the con t inuous p h a s e . H o w e v e r , a s the fluid m o v e s , i t s d e f o r m a t i o n c a u s e s t r a n s l a t i o n a l and r o t a t i o n a l nnotions of o t h e r p a r t i c l e s in the ne ighborhood . S ince t h e p a r t i c l e s a r e m o r e r e s i s t a n t to d e f o r m a t i o n than i s t h e fluid, t he p a r t i c l e s i m p o s e a s y s t e m of f o r c e s t ha t r e a c t upon the fluid. As a r e s u l t of add i t iona l s t r e s s e s , the o r i g i n a l p a r t i c l e s e e s an i n c r e a s e in the r e s i s t a n c e to i t s m o t i o n , wh ich a p p e a r s to it a s a r i s i n g f r o m an i n c r e a s e of the v i s c o s i t y . C o n s e q u e n t l y , in an a n a l y s i s of the m o t i o n of the s u s p e n d e d p a r t i c l e s , t he m i x t u r e v i s c o s i t y should be u s e d .

In the p r e s e n t a n a l y s i s , we have ex tended T a y l o r ' s c o r r e l a t i o n for the m i x t u r e v i s c o s i t y a long the R o s c o e - t y p e p o w e r r e l a t i o n b a s e d on t h e m a x i m u m pack ing Q^drn- T h u s ,

,, / cy, ^-2•5Q'dm(P'd+o•4^ic)/(^^d+^^c)

^ c V ^dm/

The m a x i m u m pack ing Q'dm for s o l i d - o r l i q u i d - p a r t i c l e s y s t e m s r a n g e s f r o m 0.5 to 0 .74. H o w e v e r , Q'dm. = 0.62 suff ices for m o s t p r a c t i c a l c a s e s . F o r a bubbly flow, the t h e o r e t i c a l va lue of <^^Yn ^^^ ^^ m u c h h i g h e r . T h e r e f o r e , by c o n s i d e r i n g the s t a n d a r d r a n g e of i n t e r e s t of the void f r a c t i o n o d "^ 0 .35 for deve loped flow and c d h i g h e r t han 0.3 5 in deve lop ing f lows , and in v iew of t h e p o s s i b l e foam r e g i m e , we m a y t a k e O!^^^ = ^ 1 . A s i m i l a r a r g u m e n t can be u s e d for a d r o p l e t flow.

F i g u r e 3 c o m p a r e s the p r o p o s a l for p r e s e n t m i x t u r e v i s c o s i t y wi th the v a r i o u s ex i s t i ng m o d e l s for s o l i d - p a r t i c l e s y s t e m s . Inc lud ing the effect of the v i s c o s i t y of the d i s p e r s e d p h a s e in the c o r r e l a t i o n g ives the newly deve loped m o d e l an a d v a n t a g e o v e r the conven t iona l c o r r e l a t i o n s , b e c a u s e it is not l i m i t e d to p a r t i c u l a t e f lows , but can a l s o be appl ied to d r o p l e t and bubbly f lows.

Now le t us a s s u m e tha t in the v i s c o u s r e g i m e a c o m p l e t e s i m i l a r i t y e x i s t s b e t w e e n CQOO b a s e d on NRQOO and Cj) b a s e d on N R Q ' so tha t C D h a s exac t l y t h e s a m e funct ional f o r m in t e r m s of NRe a s Cjyx in t e r m s of NRe<» given by Eq . A .4 . T h e n , by c o n s i d e r i n g the two a s y m p t o t i c c a s e s of NRg -• 0 and N R Q -* 00 and i n t e r p o l a t i n g b e t w e e n t h e m , we obta in , f r o m E q . 19,

v_ , 1 + O.lN^'l^.

— v - - d ' ^^-d/ J ^ 0 .1N0-" [f(c. J ]6A ^ - (1 - , ) ^ / ^ f ( c . d ) : — T T T X ^ T ^ r ^ ^ ' (21) Reo°'

w h e r e

Ha^) . (1 - ^^V%JV^^. (22)

E

uuuv

1000

100

10

1

;

: -

r

-

-

~

^

© ® ® ® ®

1 1 1 BRINKMAN (0^^=0.62)"

1

ROSCOE (Ci=L35, 0^^ = 0.62)'°

LANDELelal"

FRANKEL and ACRIVOS*"

EILERS (aj^=0.62)^'

® THOMAS (adm = 0.62)"

^ 1 1 1

®

/

/

1 E

_

{III 1

1 1 1® :

® -

/ 1

^ ^ PRESENT MODEL

(WITH/ij»,i^) -

1 1 0.2 0.4 0.6

"d/"dm

0.8 1.0 1.2

Fig. 3

Mixture-viscosity Model. ANL Neg. No. 900-76-441 Rev. 1.

C o n s e q u e n t l y , t he dr i f t ve loc i t y , wh ich is r e l a t e d to the r e l a t i v e ve loc i t y by V^j = (l _ a^)vj . , can be g iven by

1 + Mr*) Vdj - v , „ ( l - ^ d ) ' " f ( ^ d ) , , ^ ( , . ) ^ , ( , ^ ) j . / . . (23)

where ili(rd) = 0 .55[ ( l+0 .08ry) ' ' ^ ' ' - l]""' '^ fo r thev i scous regime, and r J i s the non-dimensional radius given by r^ = rd[pcgAp/|ic]^ - Note that Nj gce in Eq. 21 has been replaced by a new function M using the identity NRgoo = 2r^v*co and Eq. A.7.

The s imi lar i ty c r i te r ion given by Cj-)(N-Og) = ^D«>^''^Re) '^^^^ the Reynolds number based on the mixture viscosi ty is f i rs t introduced for the so l id-par t ic le sys tem in the Stokes regime.^ ' '* In this case , the velocity ra t io B reduces to B = (l - oi^\^^j\i.^^, which is the limiting case of Eq. 21 with Nj goo "^ 1 or ^d « 1. However, because of the use of the general ized drag law and the mix ture -v i scos i ty model in the analysis , the present theory is not limited to a so l id-par t ic le sys tem or to the Stokes reg ime.

Fo r so l id-par t ic le sy s t ems , we assume that the t rans i t ion from the viscous regime to Newton's regime occurs at the same radius as in the single-par t ic le sys tem, and that the drag coefficient C D is a continuous function. Then, for Newton's reg ime given by rd ^ 34.65, we obtain

18.67 Vdj = v^cod - Q^d)'-'f(Q'd) 1 + I7.67[f(ad)]^/^

(24)

w h e r e f(Q'd) i s g iven by Eq . 22.

F i g u r e 4 c o m p a r e s t h e above t h e o r e t i c a l r e s u l t s given by E q s . 23 and 24 wi th t h e ennp i r i ca l c o r r e l a t i o n for s o l i d - p a r t i c u l a t e s y s t e m s . ^ An a g r e e m e n t at r e l a t i v e l y low v o l u m e t r i c c o n c e n t r a t i o n s is e x c e l l e n t at a l l R e y n o l d s -n u m b e r r e g i o n s . At v e r y high v a l u e s oia^, the p r e s e n t t h e o r y p r e d i c t s m u c h l o w e r dr i f t v e l o c i t i e s t h a n the R i c h a r d s o n - Z a k i co r re l a t ion .^^ H o w e v e r , t he o r i g i n a l e x p e r i m e n t a l da t a of R i c h a r d s o n and Z a k i a l s o i n d i c a t e t h i s t r e n d , wh ich is p r e d i c t e d by the p r e s e n t t h e o r y , as shown in F i g . 5.

e =>" 0.

0.01

PRESENT THEORY (WITH (1^^ = 0.62)

RICHARDSON AND ZAKIS EMPIRICAL CORRELATION

0.1 0.2 0.3 OH

0.4 0.5

1.0

0.6

Fig. 4. Comparison of Predicted Results with Richardson and Zaki's Empirical Correlations for Solid-particulate Flow Systems. ANL Neg. No. 900-76-442.

1 1 1 1 1 1

° RICHARDSON AND ZAKl'S DATA" (6.2cm ID, r=0.3l75cm) j

RICHARDSON AND ZAKl'S-^.. / EMPIRICAL C O R R E L A T I O N " ^ - ^

9/

i PRESENT THEORY (aj- = 0.62)-./ l/ /Sn

/COD/

/' °j 1 °j

1 1 / 1 / 1 1 1

1 ly

/ -

-

1 1

0.1 —

0.1 0.2 0.4 0.6 0.8 1.0

VOID FRACTION OF CONTINUOUS PHASE, ( l -a^)

Fig. 5. Comparison of Present Model with Experimental Data for Solid Particles. ANL Neg. No. 900-76-444 Rev. 1.

In the d i s t o r t e d - f l u i d - p a r t i c l e r e g i m e , the s i n g l e - p a r t i c l e d r a g coeff i c ien t d e p e n d s only on the p a r t i c l e r a d i u s and fluid p r o p e r t i e s and not on t h e v e l o c i t y o r the v i s c o s i t y ; i . e . , CDO? = ( 4 / 3 ) r d ' / g A p / a . T h u s , for a p a r t i c l e of a fixed d i a m e t e r , CDOO b e c o m e s c o n s t a n t . In c o n s i d e r i n g the d r a g coef f ic ien t for a m u l t i p a r t i c l e s y s t e m wi th t h e s a m e r a d i u s , we m u s t t a k e into accoun t t h e r e s t r i c t i o n s i m p o s e d by t h e e x i s t e n c e of o t h e r p a r t i c l e s on t h e flow f ie ld .

B e c a u s e of the r a n d o m c h a r a c t e r i s t i c of t h e t u r b u l e n t edd ie s and p a r t i c l e o s c i l l a t i o n s , a p a r t i c l e s e e s the i n c r e a s e d d r a g due to o t h e r p a r t i c l e s in e s s e n t i a l l y the s a m e way a s in N e w t o n ' s r e g i m e for a s o l i d - p a r t i c l e s y s t e m w h e r e CDOO i s c o n s t a n t u n d e r a t u r b u l e n t flow condi t ion .

H e n c e we p o s t u l a t e t ha t , r e g a r d l e s s of the d i f f e r e n c e s in CQCO in t h e s e r e g i m e s , the effect of i n c r e a s e d d r a g can be p r e d i c t e d by the s a m e e x p r e s s i o n . U n d e r th i s a s s u m p t i o n , Eq . 24 m a y a l s o be used for the d i s t o r t e d - p a r t i c l e r e g i m e wi th t h e a p p r o p r i a t e Vj-oo- Subs t i t u t ing t h e def in i t ion of f(Q'd) in to E q . 24, the r e s u l t s can be f u r t h e r s imp l i f i ed to

Vdj =- V r ~ ^ <

(1 - c ^ d ) ' * " ' ^ ^ c » ^ ^ d

(1 - c^d)'; M-c "- ^ d (^^)

(1 - ^ d ) ' - " ' ^ ^ d » ^ ^ c

for the d i s t o r t e d - f l u i d - p a r t i c l e r e g i m e |vj.ool = v 2(gaAp/p^)° '^^. The above c r i t e r i o n is a p p l i c a b l e for Njj ^ 0.11 (l + '|')/i|' . F o r a d i s t o r t e d - b u b b l y flow the f i r s t e x p r e s s i o n u n d e r the condi t ion ij, , » \i^ i s a p p l i c a b l e , wh ich is v e r y c l o s e to the s e m i e m p i r i c a l c o r r e l a t i o n of Z u b e r and F i n d l e y ; i . e . , V^j: = Vj.oo(l - Q'd)^'^- -^or cx^ « o'dm- the above e x p r e s s i o n wi th p,d - - M'c ^^^ ^® used for N e w t o n ' s r e g i m e wi th \v^„\ = 2.43 Vgr^Ap/p^, .

As the r a d i u s of the fluid p a r t i c l e is f u r t h e r i n c r e a s e d , the w a k e and bubble b o u n d a r y l a y e r can o v e r l a p due to t h e f o r m a t i o n of l a r g e w a k e r e g i o n s . In o the r w o r d s , a p a r t i c l e can in f luence both the s u r r o u n d i n g fluid and o t h e r p a r t i c l e s d i r e c t l y . H e n c e the e n t r a i n m e n t of a p a r t i c l e in a wake of o t h e r p a r t i c l e s b e c o m e s p o s s i b l e . Th i s flow r e g i m e i s known as the c h u r n - t u r b u l e n t -flow r e g i m e and is c o m m o n l y o b s e r v e d in bubbly f lows. In th i s flow reg inne , a t yp i ca l p a r t i c l e m o v e s wi th r e s p e c t to the a v e r a g e v o l u m e t r i c flux j r a t h e r than at the a v e r a g e ve loc i ty of a cont inuous p h a s e . H e n c e , the r e f e r e n c e ve loc i ty in the def in i t ions of the d r a g coeff ic ient and the Reyno lds n u m b e r should be the dr i f t ve loc i ty r a t h e r than the r e l a t i v e ve loc i t y .

In a c h u r n - t u r b u l e n t - f l o w r e g i m e , s o m e p a r t i c l e s should have r e a c h e d the d i s t o r t i o n l i m i t c o r r e s p o n d i n g to the c a p - b u b b l e t r a n s i t i o n o r the d r o p l e t d i s i n t e g r a t i o n . Th i s l i m i t can be given by the W e b e r - n u m b e r c r i t e r i o n b a s e d on the dr i f t ve loc i ty a s 2pcV^-rd/cr = 8 (bubble) o r 12 (d rop le t ) . Due to the e n t r a i n m e n t of p a r t i c l e s in a wake of o t h e r p a r t i c l e s and the c o a l e s c e n c e and d i s i n t e g r a t i o n c a u s e d by the t u r b u l e n c e , the a v e r a g e mo t ion of the d i s p e r s e d p h a s e is m a i n l y g o v e r n e d by t h o s e p a r t i c l e s that s a t i s fy the W e b e r - n u m b e r c r i t e r i o n . Thus for a c h u r n - t u r b u l e n t flow, we have F Q = -CQPj ,Vj j - |Vj j |nr^ /2 wi th C D = 8 / 3 . T h e r e f o r e , in a s t a n d a r d f o r m ,

C Q

^D = --2-Pc^rl^rl"4' ( 6)

where C D = 8(1 - c d) /3- Hence, by balancing the drag force with the p r e s s u r e and gravity forces , we obtain

^dj -

^fl I /ogAp

or 1.57

rgApx'^'' Pc - Pd

Ap (1 - .^Yi^

v /2 agAp\ Pc - Pd

Pl Ap (27)

In the exact express ion for Vdi, the proportionali ty constant v 2 is applicable for bubbly flows and 1.57 for droplet flows. However, in view of the uncertainty in predicting the drag coefficient, this difference as well as the effect of the void fraction may be neglected. The above resul t is consistent with the study by Zuber and Findley for bubbly flows.^° The present resu l t s for the drift velocity a re summar ized in Table I by considering four different cases : bubbles in liquid, droplets in liquid, droplets in gas, and solid par t ic les in gas or liquid.

T A B L E I. S u m m a r y of Drif t V e l o c i t y V j ; in Inf in i te M e d i a

Bubble in L iqu id D r o p l e t in L iqu id D r o p l e t in G a s Sol id P a r t i c l e

t^c/*^m (1 - « J ~(1 - « H ) ' ~ ( l - o . ) ^ - ^ ^ ( 1 - H ) " '

Stokes R e g i m e -^ ^ (1 - « J ) ^ 2 g ^ P ^ d

9 2 g(Pc

9 P j

M'c - d d - - d ) '

U n d i s t o r t e d -p a r t i c l e R e g i m e

1 + ill N , 5 0 . 1 1 ^

D i s t o r t e d -p a r t i c l e R e g i m e

lO.Sl^c l^c (J _ ^ )2 ^ ^ ( 1 + •) Pc^- Pd

Pc'^d ^x 1 + l l l i ^ (1 - ff.

.1 6 /7 ^ P

w h e r e % = 0 .55[(1 + O.OSr^^)'' ^ - l ] " ' "

, n P c - Pd yi ( T T ^ - d -~Tf

F o r Ne-wton's R e g i m e ( r * 2 34.67) ,

i|t = 17.67

n = 1.75 n=- 2.0 n = 2 .25

C h u r n - t u r b u l e n t ,JZ R e g i m e

agAp' y-2 CTgApV'' Pc - Pd Ap

IV. LOCAL DRIFT VELOCITY OF DISPERSED FLOW IN CONFINED CHANNEL

The constitutive equation for the drift velocity for a mul t ipar t ic le sys tem in an infinite medium has been studied in the preceding section. We now study the wall effect on the local drift velocity in a channel with uniform c r o s s - s e c t i o n a l a r ea . Under the s teady-s ta te condition without phase change {T]^ - O) and with negligible t r a n s v e r s e p r e s s u r e gradient, i .e . , Pc = Pd = Pm(z), the axial component of the local momentum equation for phase k can be reduced from Eq. 11 to

^Pm 0 = - o ' k - ^ - QlcMTk - QOtPkgz + Mik, (28)

where Mrk si d Mik a r e the axial forces assoc ia ted with the t r a n s v e r s e s t r e s s gradient and the interfacial d rag , as shown in Eq. 11, Here we have taken gz ^ 0 such that v > 0 when the flow is upward. Consequently we obtain

^Pm\ Mic = ^clPcgz + ~ r ^ y + MTCQ'C

and (29)

bpm\ Mid = »dlPdgz + —^—) + MTd«d-

However, from the interfacia l - force balance, we have

Mic + Mid = 0. (30)

Thus by eliminating the interfacial forces from the above two equations, we get the local z components of the mix tu re -momentum equation as

—^ = -Pmgz - Mxm. (31)

where Mxm ^^ ^^^ force associa ted with mix ture t r a n s v e r s e s t r e s s gradient and given by Mrm = o^dMtd + (1 - ad)MTc-

F r o m Eq. 29 and 31, we obtain

Mid = -ad[(l - o'd)gz(Pc - Pd) + (Mxm - Mrd)]. (32)

However, inasmuch as the interfacial force Mid is re la ted to the drag force by F D = MidVd/ck'di Eqs. 15 and 32 can be solved for the re la t ive velocity as

V r V 8 i-d

' ' ' '•' " T c ^ f ^ ^ c - Pd)gz(l - Qfd) + (Mxm - Mxd)]. (33)

By introducing the nondimensional groups

g * = gz(Pc - Pd)/(Apg)

and

Mx = (Mxm - Mxd)/(Apg)

(34)

where Ap = |pc - Pd|, we obtain, from Eqs. 18, 19, and 33,

C D j N R e J = C D ( N R e ) ( ^ ) / | g * ( l " c^) + M*|. (35)

By using the same s imi lar i ty hypothesis as used for sys tems in infinite media , we can find the express ion for the velocity ra t io for a channel flow. The resu l t s a r e summar ized below:

Undis tor ted-par t ic le Regime:

V dj= l ^ r J ( l - ^ d ) [ g * ( l - d ) ^ M * ] I ^ ^ ^ ^ ^ f ^ ^ (36)

where f(o'd) = (P-c/P-m) lg*( 1 " °'d.) + Mf f' and t ( r^) is given by

,4/7 0.75

Hrl)

0.55[(1 + 0,08r3^) - 1 ] ; r * < 34.65;

(37)

17,67; r^ ^ 34.65.

Dis to r ted-par t i c le Regime:

18.67 V , j = | v , J ( l - a , ) - | g . ( l - a,} . M ^ l ^ - ^ ^ ^ ^ ^ ^ p ^ - p

Churn-turbulent-f low Regime:

' V P U | g * ( l - a d ) + M * l ° - "

(38)

(39)

If the effect of the wall can be neglected and the flow is ve r t i ca l , g* = 1 and Mj = 0; hence, we have approximately

v,3 = vz(^)"*, ,40,

The express ion for the s ingle-par t ic le t e rmina l velocity Vroo for the c o r r e sponding flow reg imes appears in the appendix.

The above express ions indicate that the local drift velocity in the axial direct ion depends on the difference between the t r a n s v e r s e s t r e s s gradient of each phase; i .e . , Mr = (1 - Q?(J)(MTC - MTd)/(^Pg)- To take this effect into account requ i res additional information concerning the turbulent s t ruc ture of both the continuous and d ispersed phases .

V. ONE-DIMENSIONAL DRIFT-FLUX MODEL

Averaging over the c r o s s - s e c t i o n a l a r e a is useful for complicated engineering problems involving fluid flow and heat t r ans fe r , since field equations can be reduced to quas i -one-dimensional fo rms . By a r ea averaging, the information on changes of var iab les in the direct ion normal to the main flow within a channel is basical ly lost; therefore , the t rans fe r of momentum and energy between the wall and the fluid should be expressed by empi r ica l cor re la t ions or by simplified models . The rat ional approach to obtain a one-dimensional model is to integrate the th ree-d imens iona l model over a c r o s s -sectional a rea and then to introduce proper mean values .

A simple a r ea average over the c ros s - sec t i ona l a r ea A is defined by

<F> = ^ / F dA, (41)

^ A

and the void-fraction-weighted mean value is given by

« F k » = <c^Fk>/<c^>. (42) In the subsequent ana lys is , the density of each phase Pd and Pc within

any c ross - sec t iona l a rea is considered to be uniform, so that Pk = ((Pk)). Fo r mos t p rac t ica l two-phase flow p rob lems , this assumption is valid since the t r a n s v e r s e p r e s s u r e gradient within a channel is re la t ively small . The detailed analysis without this approximation appears in Ref. 24. Under the above simplifying assumption, the average mixture density is given by

<Pm> = <ad)Pd + (l - <ad»Pc. (43)

The a x i a l c o m p o n e n t of the we igh ted m e a n v e l o c i t y of p h a s e k i s

<QlcVk> <Jk> « v k »

<°'k> <QTc> ' (44)

w h e r e the s c a l a r e x p r e s s i o n of the v e l o c i t y c o r r e s p o n d s to the a x i a l c o m p o n e n t of the v e c t o r . Then the m i x t u r e v e l o c i t y i s def ined by

v m = "^ "^ = [ < a d > P d « v d » + (1 - < a d > ) P c « V c » ] / , <Pm>

and the v o l u m e t r i c flux i s g iven by

<j> = <Jd>+ <Jc> = < c . d > « v d » + (1 - < « d » « v c » .

The m e a n m i x t u r e en tha lpy a l s o should be w e i g h t e d by the d e n s i t y ; t h u s ,

(45)

(46)

h m = <Pm>

= [ < a d > P d « h d » + (1 - < c ^ d » P c « h c » ] / . (47)

The a p p r o p r i a t e m e a n dr i f t v e l o c i t y i s def ined by

Vdj = « v d » - <j> = (1 - < c ^ d » ( « v d » - « v c » ) . (48)

The e x p e r i m e n t a l d e t e r m i n a t i o n of the a b o v e - d e f i n e d dr i f t v e l o c i t y i s p o s s i b l e if the v o l u m e flow r a t e of e a c h p h a s e , Qk, and the m e a n void f r a c t i o n (o^d) a r e m e a s u r e d . T h i s i s b e c a u s e Eq . 48 c a n be t r a n s f o r m e d in to

Vdj iJd>

- (<Jd>+ <Jc)) . (49)

w h e r e <jk) i s g iven by ( jk ) = Q k M - F u r t h e r m o r e , the p r e s e n t def in i t ion of the dr i f t v e l o c i t y can a l s o be u s e d for a n n u l a r t w o - p h a s e flov/s. Unde r the def in i t ions of v a r i o u s v e l o c i t y f i e lds we ob ta in s e v e r a l i m p o r t a n t r e l a t i o n s , such a s

« v d » = Vm + Pc -

<Pm> Vdj,

« V e » ' m <°'d> Pd -

1 - <cvd> <Pm> ^J '

(50)

and

, . _ , <ad>(Pc - P d ) -<J) = Vm + -. r Vdj .

<Pm> (51)

In the d r i f t - f l ux f o r m u l a t i o n , a p r o b l e m is so lved for (o^d) ^^"^ ^ m wi th a g iven c o n s t i t u t i v e r e l a t i o n for Vdj . Thus Eq . 50 c a n be u s e d to r e c o v e r a so lu t ion for the v e l o c i t y of e a c h p h a s e a f t e r a p r o b l e m i s so lved .

By a r e a - a v e r a g i n g E q s . 1-4 and u s i n g the v a r i o u s m e a n v a l u e s , we ob ta in :

M i x t u r e Cont inu i ty E q u a t i o n :

&<Pm> b , / , _ , - ^ 7 - + ^ « P m ) v m ) = 0. (52)

Cont inu i ty E q u a t i o n for D i s p e r s e d P h a s e :

25<o'd>Pd b ., > _ . / ^ V b /<° 'd>PdPc-+ — « ° ' d > P d V m ) = < r d ) - — i / - \ ^ d j bt bz ^z \ <Pm>

(53)

M i x t u r e M o m e n t u m E q u a t i o n :

^<Pm>Vm b ,/ v-2 X b / \ , & / ^ ^ . T \ / „ \ TT + T - V m = - r - +-r-<T^zz + Tzz> - <Pm>gz bt bz Oz Oz

• m / v_ |_ I b ^ V m l v m l - ^

<Q^d>PdP " v^ (1 - W » ^J

bz XcOV(akPkVkVk)- (54)

M i x t u r e E n t h a l p y - e n e r g y E q u a t i o n :

- + i : 7 « P m > h m V m ) = - r r < q + qT> + bt _b_ bz

_b_ bz

q w l h b <«d>PdP<

^^ <Pm> AhdcVdj

_b_ bz

<Q'd>PdP<

<Pm) AhdcVdj I^IcOV(akPkhkVk) +

^<Pm> bt

Vm + <°'d>(Pc - P d ) -

<Pm> Vdj

!^<Pm> bz + <$i^> nn'

(55)

Here T^Z + T J ^ denotes the normal conaponents of the s t r e s s tensor in the axial direct ion and Ahdc is the enthalpy difference between phases ; thus, ^ d c = {(hd)) - ((he)). The covar iance t e r m s r ep re sen t the difference between the average of a product and the product of the average of two var iables such that COV(akPk'l'kVk) ~ ^<^kPk'l'k(vk " ((vk)))). If the profile of ei ther tk °^ Vk is flat, then the covar iance t e r m reduces to zero. The t e r m represen ted by fm(Pm)vm |vm |/(2D) in Eq. 54 is the two-phase frictional p r e s s u r e drop. We note h e r e that the effects of the m a s s , momentum, and energy diffusion a s sociated with the re la t ive motion between phases appear explicitly in the drift-flux formulation, since the convective t e r m s on the left-hand side of the field equations a r e expressed in t e r m s of the mix ture velocity. These effects of diffusions in the present formulation a r e expressed in t e r m s of the drift velocity of the d i spe r sed phase Vdj. This may be formulated in a functional form as

Vdj = Vdj((ad). <Pm). gz. Vm. etc.) . (56)

To take into account the m a s s t rans fe r a c r o s s the in ter faces , a constitutive equation for (Fd) should also be given. In a functional form, this phase-change constitutive equation may be wri t ten as

(Fd) - ( rd>\(«d) , (Pm) . Vm, — ^ 7 - . e tc . j . (57)

The above formulation can be extended to nondispersed two-phase flows, such as an annular flow, provided a proper constitutive relat ion for a drift velocity of one of the phases is given.

VI. ONE-DIMENSIONAL DRIFT VELOCITY

A. Dispersed Two-phase Flow

To obtain a kinematic constitutive equation for the one-dimensional drift-flux model , we must average the local drift velocity over the channel c r o s s section. The constitutive relat ion for the local drift velocity Vdj in a confined channel was developed in Sec. IV. Now we shall re la te this to the mean drift velocity Vdj defined by Eq. 48.

F r o m Eqs . 42 and 48,

V d i = C - ^ ^ - . > = «v,p>.(c.-», (58)

w^here

« V d j » = <cvdVdj)/(c^d> (59)

and V

(ofdj) * o ^ / v - \ - (60)

(o-dXl)

The s e c o n d t e r m on t h e r i g h t - h a n d s ide of Eq . 58 i s a c o v a r i a n c e b e t w e e n the c o n c e n t r a t i o n p ro f i l e and the v o l u m e t r i c flux p r o f i l e ; thus it c an a l s o be e x p r e s s e d a s COV(Qfdj)/(Q?d). The f a c t o r CQ, which h a s b e e n u s e d for bubb ly o r s lug f lows by s e v e r a l authors,^" '^^ '^^ i s known a s a d i s t r i b u t i o n p a r a m e t e r . The i n v e r s e of t h i s p a r a m e t e r w a s a l s o u s e d in the e a r l y w o r k of Bankoff.^^ P h y s i c a l l y , t h i s effect a r i s e s f r o m the fact tha t the d i s p e r s e d p h a s e is l o c a l l y t r a n s p o r t e d wi th the dr i f t v e l o c i t y Vdi wi th r e s p e c t to l o c a l v o l u m e t r i c flux j and not to the a v e r a g e v o l u m e t r i c flux ( j ) . F o r e x a m p l e , if the d i s p e r s e d p h a s e i s m o r e c o n c e n t r a t e d in the h i g h e r - f l u x r e g i o n , t hen the m e a n t r a n s p o r t of the d i s p e r s e d p h a s e i s p r o m o t e d by h i g h e r l o c a l j .

The va lue of Cg c a n be d e t e r m i n e d f r o m a s s u m e d p r o f i l e s of the vo id f r a c t i o n ad and to t a l v o l u m e t r i c fluxj,^° o r f r o m e x p e r i m e n t a l da ta . By a s suming p o w e r - l a w p r o f i l e s in a p ipe for j and a^, we h a v e

j/jo = 1 - (R /Rw)"^ ;

(61) Qd " Q'dW

^"^ = 1 - (R/Rw)^. "'do - Q'dW

w h e r e JQ, Qfdo, Q?dW' ^> and R w a r e , r e s p e c t i v e l y , the va lue of j and ex at the c e n t e r , the void f r a c t i o n at the w a l l , r a d i a l d i s t a n c e , and the r a d i u s of a p ipe . By subs t i t u t i ng t h e s e p r o f i l e s into the def in i t ion of Cg given by Eq . 60, we ob ta in

The d i s t r i b u t i o n p a r a m e t e r b a s e d on the above a s s u m e d p r o f i l e s is f u r t h e r d i s c u s s e d in Ref. 11 .

Now Eq. 58 c a n be t r a n s f o r m e d to

« v d » = : ^ = C o ( j ) + ( ( V d j » , (63) (ad>

w h e r e ( (vd) ) and (j > a r e e a s i l y o b t a i n a b l e p a r a m e t e r s in e x p e r i m e n t s , p a r t i c u l a r l y u n d e r an a d i a b a t i c condi t ion . T h e r e f o r e , t h i s equa t ion s u g g e s t s a p lo t of the m e a n v e l o c i t y ( (vd) ) v e r s u s the a v e r a g e v o l u m e t r i c flux ( j ) . V a r i o u s e x p e r i m e n t a l da ta u s i n g th i s m e a n v e l o c i t y - f l u x p l a n e a r e in F i g s . 6 -12 . If the c o n c e n t r a t i o n p ro f i l e i s u n i f o r m a c r o s s the c h a n n e l , then the va lue of the d i s t r i b u t i o n p a r a m e t e r i s equa l to uni ty . In add i t i on , if the effect of the l o c a l dr i f t ( (Vdj)) i s neg l ig ib ly s m a l l , t hen the flow b e c o m e s e s s e n t i a l l y h o m o g e n e o u s . In th i s c a s e , the r e l a t i o n b e t w e e n t h e m e a n v e l o c i t y and flux r e d u c e s to a s t r a i g h t l ine t h r o u g h the o r i g i n a t an ang le of 45°. The d e v i a t i o n of the e x p e r i m e n t a l da ta f r o m t h i s h o m o g e n e o u s flow l ine show^s the m a g n i t u d e of the dr i f t of the d i s p e r s e d p h a s e wi th r e s p e c t to t h e v o l u m e c e n t e r of the m i x t u r e .

An i m p o r t a n t c h a r a c t e r i s t i c of such a p lo t is t ha t , for t w o - p h a s e flow r e g i m e s wi th fully d e v e l o p e d void and v e l o c i t y p r o f i l e s , the da ta po in t s c l u s t e r a r o u n d a s t r a i g h t l ine ( s e e F i g s . 6 -12) . T h i s t r e n d i s p a r t i c u l a r l y p r o n o u n c e d when the l o c a l d r i f t v e l o c i t y i s c o n s t a n t o r neg l ig ib ly s m a l l . H e n c e , for a g iven flow r e g i m e , the v a l u e of the d i s t r i b u t i o n p a r a m e t e r CQ m a y be o b t a i n e d f r o m the s lope of t h e s e l i n e s , w h e r e a s the i n t e r c e p t of th i s l ine wi th the m e a n ve loc i t y a x i s c a n be i n t e r p r e t e d a s the w e i g h t e d m e a n l o c a l dr i f t v e loc i t y , ( (Vdj) ) . The e x t e n s i v e s tudy by Z u b e r et a l . ' ^ shows tha t Cg d e p e n d s on p r e s s u r e , c h a n n e l g e o m e t r y , and p e r h a p s flow r a t e . An i m p o r t a n t effect of s u b -coo led bo i l ing and deve lop ing void p ro f i l e on the d i s t r i b u t i o n p a r a m e t e r h a s a l s o b e e n no ted by H a n c o x and Nicoll.^^

0 02 0.4 06 08 10 1.2

TOTAL VOLUMETRIC FLOW, m/s

14

jn 6 E

>-"

o

o

1 \

o ZUBER

D=l cm

5.8 atm (0.58 MPa) C„=LI3

- « V g , » FOR CHURN FLOW

2 3 4 5 6 7

TOTAL VOLUMETRIC FLUX, m/s

Fig. 6. Experimental Data for High-pressure System.28 ANL Neg. No. 900-77-187 Rev. 1.

Fig. 7. Freon-22 Data in Boiling Flow.H ANL Neg. No. 900-77-181 Rev. 1.

II

10

9

4" 8 E

6' 7 V

V

^ 5

S ^ 4

S 3

2

—

—

—

—

:

—

^

1 1 1 1

o THOME

064 X 59 cm CHANNEL

latm (0 1 MPa)

°o

/ ^ O

1 1 1

oP

y ' C,.|3

^ « V g , » FOR CHURN FLOW

1 1 1 1 1 1

7 / -

—

—

—

~,

\

1 0 1 2 3 4 5 6 7 8


Fig. 8. Fully Developed N2-NaK Flow.29 ANL Neg. No. 900-77-177 Rev. 1.

15

14

13

12

. II

"^ 10 A

•i . 8 V

o S 6

O SMISSAERT

D=70 cm I Qtm (0 I MPa) <] ,>, 0-305 cm/s

^«Vg,» FOR CHURN FLOW

0 2 4 6 8 10 I


Fig. 9. Fully Developed Air-Water Flow Data.30 ANL Neg. No. 900-77-182 Rev. 1.

20

A o V —. A

3 10 LLJ

04 -

O SMISSAERT

D=5I cm I aim (OlMPo) O^ <),>, 0-30 5 cm/s

— « V „ , » FOR SLUG FLOW

O . « V g j » FOR CHURN FLOW

1 0 2 04 06 08 10

TOTAL VOLUMETRIC FLUX <)>, m/s

12

25

20

«, 15 e h 10

^ 05

LOCI

TY

o

> 5 - 0 5

1— i/t

-10

-15

-20

1 1 1 1 — OPETRICK '

" 0 = 5 cm 41 atm (4 IMPa)

— —

— /

/ rfiwr

- ^»^

<^ - J^ _ ^ DOWN FLOW

1 1 1 1

1 1 1 y

I -8^Co=ll5

/ ' / / UP FLOW _\

/ — __«V » FOR CHURN FLOW

—

—

1 1 1 1 -25 -20 -15 -10 -05 0 05 10 15 20 25


Fig. 10. Fully Developed N2-Mercury Flow Data.30 Fig. 11. Experimental Data for Cocurrent Upflow and ANL Neg. No. 900-77-176 Rev. 1. CocurrentDownflowof Steam-Water System,

ANL Neg. No. 900-77-180 Rev. 1.

25

20

15

10

t n

e 5

,"=' 0

E -5 S ^ -10

5 -15 n

-20

-25

-30

1 e

—

-

—

—

—

-

-

^

1 1 1 1 32

oBERGONZOLI

0 = 1 3 cm

l - 2 a l m ( 0 1 - 0 2 MPa)

/

/

/

/ /

"^ DOWN FLOW

1 1 1 1

' ' ' < /

c^sr Co=ii^

^ fp "~

y' UP FLOW

-

—

—

-

1 1 1

Fig. 12

Experimental Data for Cocurrent Upflow and Cocurrent Downflow of Heated Santowax-R System. ANL Neg. No. 900-77-185 Rev. 2.

-30 -25 -20 -15 -10 -5 0 5 10


15 20

In the p r e s e n t s tudy , a s i m p l e c o r r e l a t i o n for the d i s t r i b u t i o n p a r a m e t e r in bubb ly- f low r e g i m e h a s b e e n d e v e l o p e d a long the above s t u d i e s . F i r s t , by c o n s i d e r i n g a fully deve loped bubbly flow, we a s s u m e d tha t Cg d e p e n d s on the d e n s i t y r a t i o Pg/pf and on the R e y n o l d s n u m b e r b a s e d on l iqu id p r o p e r t i e s , GD/[j;f, w h e r e G, D, and txf a r e the t o t a l m a s s flow r a t e , h y d r a u l i c d i a m e t e r , and the v i s c o s i t y of the l iqu id , r e s p e c t i v e l y . H e n c e ,

P_g G D '

Pf' M-f (64)

A s i n g l e - p h a s e t u r b u l e n t - f l o w p r o f i l e and the r a t i o of the m a x i m u m v e l o c i t y to m e a n v e l o c i t y give a t h e o r e t i c a l l i m i t i n g va lue of Cg a t a ^ 0 and Pg/pf -* 0, s i n c e in t h i s c a s e a l l t he bubb l e s shou ld be c o n c e n t r a t e d at the c e n t r a l r eg ion . Thus f r o m the e x p e r i m e n t a l da t a of Nikuradse^"* for a round t u b e , wh ich g i v e s the r a t i o of the m a x i m u m to m e a n v e l o c i t y , we h a v e

l i m ( a j ) <a)jo

(c^Xj) (c^)(j) = 1.393 - 0 .0155 -tnl

( ^ ) (65)

as a -* 0 and Pg/Pf -* 0 (see Fig. 13). On the other hand, as the density ra t io approaches the unity, the distr ibution p a r a m e t e r Cg should also become unity. Thus,

Cg - 1 (66)

as Pg/Pf -* 1.

, 24

20

16

12

08

04

0

1

—

—

1

1 1 1 l l l l j 1 1 1 1 l l l l l 1 1

, 1 3 9 - 0 0155 In (GD//i )

O NIKURADSE

1 1 1 1 m l 1 J 1 1 l l l l l 1 1

1 l l l l |

0 —

1 10 10

REYNOLDS NUMBER, GD//1

Fig. 13

Limiting Value of Distribution Parameter Co at Zero Void Fraction and pg/pf-* 0 Based on Single-phase Turbulent-flow Profile. ANL Neg. No. 900-77-178 Rev. 1.

B a s e d on t h e s e l i m i t s and v a r i o u s e x p e r i m e n t a l da ta in a fully d e v e l o p e d flow, the d i s t r i b u t i o n p a r a m e t e r c a n be g iven a p p r o x i m a t e l y by

Cn - Cc - (Coo - OyPg/Pf. (67)

where the density group scales the iner t ia effects of each phase in a t r a n s v e r s e void distribution. Physical ly , Eq. 67 models the tendency of the l ighter phase to migra te into a higher-veloci ty reg ime , thus result ing in a higher void concentrat ion in the cent ra l regime.^^ For a laminar flow. Coo is 2, but, due to the la rge velocity gradient , Cg is very sensi t ive to ( a ) at low void fract ions.

Over a wide range of Reynolds number , GD/jif, Eq. 65 can be approximated by Coo ^ 1-2 for a flow in a round tube. On the other hand, for a rec t angular channel, the experimental data show this value to be approximately 1.35. Thus, for a fully developed turbulent bubbly flow.

1.2 - 0.2 -v/pg/pf: round tube;

1.35 - 0.35 "^g/Pf: rec tangular channel.

(68)

F igures 14 and 15 compare the above cor re la t ion with various exper imental data. Each point in the figures r ep re sen t s anywhere from five to 150 data points. Fo r example, the original exper imental data of Smissaert ,^" shown in Fig. 9, a re presented by a single point in Fig. 14, Each point in Fig. 9 can be used to obtain a corresponding value for Cg by using the existing corre la t ion for the mean local drift velocity ((V(jj». However, in view of the strong l inear relat ion between the mean velocity of the d i spersed phase and the total flux, the average value of Cg obtained by l inear fitting has been used in F igs . 14 and 15.

In the velocity-flux plane (see Figs . 11 and 12), three operational modes can be easily identified. In the f i rs t quadrant , the flow is basical ly cocur ren t upward; therefore both the liquid and vapor phases flow in an upward direction. Examples of this mode a re shown in F igs . 6-10. In the second quadrant, the

v a p o r p h a s e i s m o v i n g u p w a r d ; h o w e v e r , t h e r e i s a ne t d o w n w a r d flow of m i x t u r e . C o n s e q u e n t l y , the flow i s c o u n t e r c u r r e n t . The c o c u r r e n t downflow o p e r a t i o n should a p p e a r in the t h i r d q u a d r a n t of the v e l o c i t y - f l u x p l a n e , a s shown in F i g s . 11 and 12. T h e s e da t a i n d i c a t e t ha t t he b a s i c c h a r a c t e r i s t i c d e s c r i b e d by Eq. 63 i s va l i d for both c o c u r r e n t up and dow^n flows wi th an i d e n t i c a l v a l u e for the d i s t r i b u t i o n p a r a m e t e r Cg. T h i s fac t d e m o n s t r a t e s the u s e f u l n e s s of c o r r e l a t i n g the d r i f t v e l o c i t y in t e r m s of the m e a n l o c a l d r i f t v e l o c i t y ((V^j; )) and Cg.

2.5

" 2 . 0

1.5 —

0.5

1 ' 1 1 1 • ZUBERt FREON-22, 0 = 1 cm

O MAURER^ FREON-22, D=lcm

A HUGHES?' WATER, 0 = 16.8 cm " ~ A SMISSAERT,'" AIR-WATER, D=5.lcm

D WALLIS?^ AIR-WATER, 0=5.1 cm • ROSEf^ AIR-WATER, 0=2.5 cm

O ADORNlf' WATER, 0 = 2.5 cm

_ • ROUHANI?' H. WATER, D=0.e cm

^ SCHWARTzt" WATER, 0=6 cm

^

PRESENT CORRELATION (C„ = l . 2 - 0 . 2 y p g / / . , )

1 1 1 1 1 1 < 1

1

~

-

1 1 0.2 0.4 0.6 0.8

A^ 1.0

2.0

1.5

1.0

1

"

—

KA

1

•

O

A

u

1 ' 1 ' 1 ' 1 St. PIERREf WATER, I.I x 4.4 cm

MARCHATERRE,'' WATER, I.I x 9.4 cm

BAKER," R-ll, 0.97 x 27 cm

T H O M E P N^-NOK, 0.64 x 5.9 cm

PRESENT CORRELATION (C„= l . 35 -0 .35yp^ /p , )

1 1 1 1 1 1 1

1

—

-

1

0.2 0.4 0.6 0.8 1.0

Fig. 14. Distribution Parameter for Fully Developed Flow in a Round Tube. ANL Neg. No. 900-77-173 Rev. 1.

Fig. 15. Distribution Parameter for Fully Developed Flow in a Rectangular Channel. ANL Neg. No. 900-77-184 Rev. 1.

I n t w o - p h a s e s y s t e m s w i t h h e a t a d d i t i o n , t h e c h a n g e of v o i d p r o f i l e s f r o m c o n c a v e t o c o n v e x c a n o c c u r a s i l l u s t r a t e d i n F i g s . 16 a n d 17. T h e c o n c a v e v o i d - f r a c t i o n p r o f i l e i s c a u s e d b y t h e w a l l n u c l e a t i o n a n d d e l a y e d t r a n s v e r s e m i g r a t i o n of b u b b l e s t o w a r d t h e c e n t e r of a c h a n n e l . U n d e r t h e s e c o n d i t i o n s , m o s t of t h e b u b b l e s a r e i n i t i a l l y l o c a t e d n e a r t h e n u c l e a t i n g w a l l . T h e c o n c a v e p r o f i l e i s p a r t i c u l a r l y p r o n o u n c e d i n t h e s u b c o o l e d b o i l i n g r e g i m e , b e c a u s e h e r e o n l y t h e w a l l - b o u n d a r y l a y e r i s h e a t e d a b o v e t h e s a t u r a t i o n t e m p e r a t u r e a n d t h e c o r e l i q u i d i s s u b c o o l e d . T h i s t e m p e r a t u r e p r o f i l e w i l l i n d u c e c o l l a p s e s of m i g r a t i n g b u b b l e s i n t h e c o r e r e g i o n a n d r e s u l t a n t l a t e n t h e a t t r a n s p o r t f r o m t h e w a l l t o t h e s u b c o o l e d l i q u i d . H o w e v e r , a s i m i l a r c o n c a v e p r o f i l e c a n a l s o b e o b t a i n e d b y i n j e c t i n g g a s i n t o f l o w i n g l i q u i d t h r o u g h a p o r o u s t u b e wal l .^^

I n t h e r e g i o n i n w h i c h v o i d s a r e s t i l l c o n c e n t r a t e d c l o s e t o t h e w a l l , t h e m e a n v e l o c i t y of v a p o r c a n b e l e s s t h a n t h e m e a n v e l o c i t y of l i q u i d , b e c a u s e t h e b u l k of l i q u i d m o v e s w i t h t h e h i g h c o r e v e l o c i t y . H e n c e , i n t h i s r e g i o n .

the data should fall below the l ine for a fully deve loped two-phase flow. This i s i l lus trated in F i g s . 18-20. If the data fall be low the homogeneous l ine , the value of Cg should be l e s s then unity. However , a s m o r e and m o r e vapor i s

0 01 02 03 04 05 06 07 08 09 10 II 12 AXIAL DISTANCE, m

Fig. 16. Axial Void Distribution with Corresponding Void Profiles at Various Stations for Steam-Water Experiment in Rectangular Channel. ANL Neg. No. 900-77-172 Rev. 1.

0 01 02 03 04 05 06 07 08 09 10 II 12 13 14 15 AXIAL DISTANCE, m

Fig. 17. Axial Void Distribution with Corresponding Void Profiles at Various Stations for Freon-22 Experiment in Round Tube. ANL Neg. No. 900-77-183 Rev. 1.

3 0

25

2 0

QC 10 o 0.

—

—

A k3

1 1 1 1 O ROUHANI ond BECKER

STEAM-HEAVY WATER

D = 0 6 cm /

19 atm (l9MPo) /

p . ^•^^gi* ""O" CHURN FLOW

-X 1 1 1

/

/ \ Co'113

1

—

—

—

5 10 15 2 0 25

TOTAL VOLUMETRIC FLUX < j > , m/s

3 0

Fig. 18

Experimental Data of Rouhani and Becker39 in Round Tube and Effect of Developing Flow due to Boiling. ANL Neg. No. 900-77-179 Rev. 1.

50

45

40

35

30

25

20

I 15 >

O St PIERRE STEAM-WATER 11 X 4 4 cm 28 I otm (2 81 MPa)

10

0 5

0 0 5 10 15 2 0 25 30 35 4 0

TOTAL VOLUMETRIC FLUX < ) > , m/s

Fig. 19

Experimental Data of St. Pierre^l in Rectangular Flow and Effect of Developing Flow due to Boiling. ANL Neg. No. 900-77-174 Rev. 1.

4 5

4 0

3 5

3 0

2 5

2 0

I 5

10

0 5

O MARCHATERRE

STEAM-WATER

I I X 9 4 cm DUCT

21 4 atm (2 14 MPa)

0 5 10 15 2 0 2 5 3 0 3 5

TOTAL VOLUMETRIC FLUX < ) > , m/s

4 0

Fig. 20

Experimental Data of Marchaterre42 in Rectangular Duct and Effect of Developing Flow due to Boiling. ANL Neg. No. 900-77-175 Rev. 1.

g e n e r a t e d a l o n g the c h a n n e l , the v o i d - f r a c t i o n p r o f i l e c h a n g e s f r o m c o n c a v e to c o n v e x a n d b e c o m e s f u l l y d e v e l o p e d . T h i s t r e n d c a n e a s i l y b e s e e n i n F i g s . 1 8 - 2 0 , i n w h i c h t h e data a p p r o a c h the f u l l y d e v e l o p e d f l o w l i n e f r o m b e l o w in t h e bulk b o i l i n g r e g i m e .

F o r a flow wi th g e n e r a t i o n of void a t t h e w a l l due to e i t h e r n u c l e a t i o n o r gas in jec t ion , the d i s t r i b u t i o n p a r a m e t e r Cg should h a v e a n e a r - z e r o v a l u e a t the beg inn ing of t h e t w o - p h a s e flow r e g i o n . T h i s c a n be a l s o s e e n f r o m the def in i t ion of Cg in Eq . 60. H e n c e , we h a v e

l i m (a>-0^° l i m

(c^j) <Q')j w (cy)(j) ( a X j )

for Tg > 0. (69)

With the i n c r e a s e in the c r o s s - s e c t i o n a l m e a n void f r a c t i o n , t he p e a k of the l o c a l void f r a c t i o n nnoves f r o m the n e a r - w a l l r e g i o n to the c e n t r a l r e g i o n , a s shown in F i g s . 16 and 17. T h i s wi l l l e ad to the i n c r e a s e in the va lue of Cg a s the void p ro f i l e d e v e l o p s .

In v iew of the b a s i c c h a r a c t e r i s t i c d e s c r i b e d above and v a r i o u s e x p e r i m e n t a l data,^^'^^''*^''*^ the fol lowing s i m p l e c o r r e l a t i o n i s p r o p o s e d :

Cg = [Ceo - (Coo - 1) ^/pJp'nil - e-''^°'^). (70)

This e x p r e s s i o n i n d i c a t e s the s ign i f i cance of the deve lop ing void p ro f i l e in the r e g i o n g iven by 0 < (o?) < 0 .25 ; beyond th i s r e g i o n , t he va lue of Cg a p p r o a c h e s r a p i d l y to tha t fo r a fully d e v e l o p e d flow ( s e e F i g . 21). H e n c e , for Tg > 0, we obta in

Cn = <

(1.2 - 0.2 ypg/Pf)( l - e-^«<°^>); r o u n d tube ;

(1 .35 - 0.35 V^Pg/Pf)(l - e ' ' ^ ^ ^ 0 ; r e c t a n g u l a r c h a n n e l .

(71)

14

, 12

10

08

6 06 -,

02

1 r

~M^° o«®^ n°°

-1° 1

f >i 1

1 ' 1 1 1 1 P,'Pr° %047

0 137

10

o DATA IN DEVELOPING REGIML

St PIERRE

MAURER

MARCHATERRE

0 0 0 5 <Pq/p,< 01

1 1 1 1 1 1

1

—

I ~ 01 02 03 04 05 06 07 08 09 10

VOID FRACTION, <a>

Fig. 21

Distribution Parameter in Developing Flow due to Boiling. (Data for the rectangular duct have been modified by a factor of 1.2/1.35 to obtain corresponding data for a round tube.) ANL Neg. No. 900-77-186 Rev. 1.

F o r m o s t d r o p l e t o r p a r t i c u l a t e f lows in the t u r b u l e n t r e g i m e , the v o l u m e t r i c flux p ro f i l e i s q u i t e flat due to the t u r b u l e n t m i x i n g and p a r t i c l e s l i p s n e a r the wa l l , which i n c r e a s e the v o l u m e t r i c flux. The c o n c e n t r a t i o n

of d i s p e r s e d p h a s e a l s o t e n d s to be u n i f o r m , excep t for w e a k peak ing n e a r the c o r e of the flow. B e c a u s e of t h e s e p r o f i l e s for j and a^j, the va lue of the d i s t r i b u t i o n p a r a m e t e r Cg i s e x p e c t e d to be c l o s e to un i ty (1.0 ^ Cg ^ 1.1). T h u s , by a s s u m i n g tha t the c o v a r i a n c e t e r m s a r e neg l ig ib ly s m a l l for d r o p l e t o r p a r t i c u l a t e f lows, we h a v e

V d j ^ ( (Vdj ) ) , (72)

in which c a s e the l oca l s l ip b e c o m e s i m p o r t a n t .

The c a l c u l a t i o n of ( (Vdj)) b a s e d on the loca l c o n s t i t u t i v e e q u a t i o n s i s the i n t e g r a l t r a n s f o r m a t i o n , Eq . 59; thus it wi l l r e q u i r e a d d i t i o n a l i n f o r m a t i o n on the void profile^^. S ince th i s p ro f i l e i s not known in g e n e r a l , we m a k e the fol lowing s impl i fy ing a p p r o x i m a t i o n s . The a v e r a g e dr i f t v e l o c i t y ((V(jj )) due to the l o c a l s l ip c a n be p r e d i c t e d by the s a m e e x p r e s s i o n a s the l o c a l c o n s t i tu t ive r e l a t i o n s g iven in Ref. 44 , p r o v i d e d the l o c a l void f r a c t i o n a^ and the n o n d i m e n s i o n a l d i f f e r ence of the s t r e s s g r a d i e n t a r e r e p l a c e d by a v e r a g e v a l u e s . T h e s e a p p r o x i m a t i o n s a r e good for flows wi th a r e l a t i v e l y flat void-f r a c t i o n p ro f i l e ; a l s o , t hey c a n be c o n s i d e r e d a c c e p t a b l e f r o m the o v e r a l l s i m p l i c i t y of the o n e - d i m e n s i o n a l m o d e l .

F o r a fully deve loped v e r t i c a l flow, the s t r e s s d i s t r i b u t i o n in the fluid and in the d i s p e r s e d p h a s e should be s i m i l a r ; thus the effect of s h e a r g r a d i e n t on the m e a n l o c a l dr i f t v e l o c i t y c a n be n e g l e c t e d . Unde r t h e s e c o n d i t i o n s we ob ta in t h e fol lowing r e s u l t s :

U n d i s t o r t e d - p a r t i c l e R e g i m e :

Vdj = (Cg - l ) ( j )

+ 10.8|ic l c

Pc^d (uim) •(l-<«d»^

t|r''/3(l + t )

1 + i|f

Pd

.(M-m) (1 - (ad))°-5

6/7 Ap (73)

w h e r e *(r^) = 0.55[(1 -I- O.OSr^^)'''"' - i f ' " for r ^ < 34.65 and i^(r|) = 17.67 for r ^ ^ 34 .65 . The l i m i t i n g c a s e of the u n d i s t o r t e d - p a r t i c l e r e g i m e i s t he S tokes r e g i m e in wh ich t h e m e a n d r i f t v e l o c i t y r e d u c e s to

V,jMC„-l) ( i> . fr |«^a-<. ,V^^ - P,

Ap (74)

D i s t o r t e d - p a r t i c l e R e g i m e (1 .75 ^ n ^ 2 .25) :

1/4 - Pc - Pd Vdj = ( C g - 1){J) + V Z ( ^ \ ( 1 - (cVd))"

Ap (75)

36

Here the value of n depends on the v iscos i t ies as seen in Eq. 25.

Churn-turbulent-flow Regime:

Here the mean mixture viscosi ty * is given by

< 0 / <a.>V2•5°'d^l(M•d+o.4|J.c)/(^Ad+^ic) -IHL = 1 1 - . (77)

^ c V Q'dm /

The value of maximum packing, o/^.^^ = 0.62, is recommended for solid part icle-f luid s y s t e m s , although it can range from 0.5 to 0.74. However , for a bubbly flow, the theore t ica l value of Q?dm ^^.n be much higher. If we consider the s tandard range of in te res t of void fraction in bubbly flow, ^'dm "^^y ^^ approximated by ffdm = 1- Hence, for a bubbly flow, the mix ture viscosi ty becomes

<M^> 1 1 - <«d>'

(78)

On the other hand, for a par t icula te flow with a low par t ic le concentrat ion, i .e . , (o'd^ " " 1' ^^^m^ can be approximated by

%B> = (1 . (ad))-^-^ (79) PC

In a horizontal flow with a complete suspension of the d ispersed phase the t r a n s v e r s e mixing, which keeps the par t i c les suspended, can significantly influence the s t r e s s gradient of each phase; thus the s t r e s s gradient effect may not be neglected. However, in view of the present state of the a r t , the assumption ((Vdj)) =» 0 may be used as a f i r s t - o r d e r approxinaation, par t icular ly in high-flux flows.

For high-flux flows, the effect of the local drift ((Vdj)) on the mean drift velocity is small in compar ison with the covariance t e r m (Cg - l ) ( j ) . Thus, by neglecting the former , we have

^J <Pm)- (Cg- l ) ( a d ) ( P c - Pd) ^

For bubbly flows, the above equation imposes a condition on applicable void-fraction ranges ; thus we should have (Pm)> (Cg - l)(Q?d)(Pc - Pd)-Here a simple c r i t e r ion for the boundary between the high- and low-flux flow can be obtained by taking the rat io of the total volumetr ic flux and the t e rmina l velocity. If this rat io is naore than 10, the flow can be considered a high-flux flow.

The other limiting case of the d i spersed two-phase flow in a confined channel is slug flow. When the volume of a bubble is very l a rge , the shape of the bubble is significantly deformed to fit the channel geometry. The d iamete r s of the bubbles become approximately that of the pipe with a thin liquid film separat ing the bubbles from the wall. The bubbles have the bullet form with a cap-shaped nose. The naotion of these bubbles in re la t ively in-viscid fluids can be studied by using a potential flow^ analysis around a sphere,'^ and the resul t is shown to ag ree with exper imental data. Thus,

V , j = 0 . 2 < j > . 0 . 3 5 ( l £ ^ y

which was originally proposed by Niklin et al.^^ and Neal.^

(81)

B. A n n u l a r T w o - p h a s e F l o w

In a n n u l a r twô-phase flow^s, the r e l a t i v e m o t i o n s betwêen p h a s e s a r e g o v e r n e d by the i n t e r f a c i a l g e o m e t r y , the b o d y - f o r c e f ie ld , and the i n t e r f a c i a l m o m e n t u m t r a n s f e r . T h e c o n s t i t u t i v e equa t ion for t h e v a p o r - d r i f t ve loc i t y in a n n u l a r t w o - p h a s e f lows h a s been d e v e l o p e d by t ak ing into a c c o u n t t h o s e n a a c r o s c o p i c ef fects of the s t r u c t u r e d twô-phase flow^s.'^ Assunaing s t e a d y -s t a t e a d i a b a t i c t w o - p h a s e a n n u l a r flow wi th c o n s t a n t s i n g l e - p h a s e p r o p e r t i e s , ^ve have the fol lowing o n e - d i n a e n s i o n a l naonaentum e q u a t i o n s for e a c h p h a s e :

dp m

dz + Pggj (c^)A (82)

and

dp m

dz + Pfg,

T-P-Pwf'^wf

A( l - (a)) ~ A( l - (a)) ' (83)

w h e r e TJ , T^f, P j , and Pwf =^re the i n t e r f a c i a l s h e a r , w a l l s h e a r , i n t e r f a c i a l w e t t e d p e r i m e t e r , and w a l l w e t t e d p e r i m e t e r , r e s p e c t i v e l y . The h y d r a u l i c d i a m e t e r and the r a t i o of wêtted p e r i m e t e r s a r e def ined by D = 4 A / P ^ and ? = P i / P w f . By a s s u m i n g tha t the filna t h i c k n e s s 6 i s s m a l l c o m p a r e d w i th D, we h a v e 4 6 / D =- 1 - (cc). On the o t h e r hand , for an a n n u l a r flow in a p i p e , 5 r e d u c e s to v t t .

The wâll s h e a r can be e x p r e s s e d t h r o u g h the f r i c t i o n f ac to r wîth a g r a v i t y - c o r r e c t i o n t e r m by T ^ £ = f-^fPf((vf))l((vf))|/2 - Apg^-S/S, w h e r e f^f can be g iven by the s t a n d a r d f r i c t i o n - f a c t o r c o r r e l a t i o n : f^f = 16 /Ref for l a m i n a r f i lm flows and f^f = 0.0791Rer°"^^ for t u r b u l e n t f lows. H e r e the l i q u i d - f i l m Reyno lds n u m b e r i s g iven by Re£ = P£l(J£)lD/M.f. S ina i l a r ly , the i n t e r f a c i a l s h e a r can be e x p r e s s e d a s T ^ = fj pg |vj.]vj./2 wi th the i n t e r f a c i a l f r i c t ion fac to r given by f = 0 .005[ l + 75(1 - (a))] for r o u g h wavy f i lms.^^

By def in i t ion , t he v a p o r - d r i f t ve loc i ty i s r e l a t e d to Vj-, i . e . , V gj

( l - (Qf))vj.; h e n c e , by e l inainat ing the p r e s s u r e g r a d i e n t frona the naonaentum e q u a t i o n s , we ob ta in for a l a m i n a r f i lm

^ g j = ± 16(a)

Pgfil

M'f<jf) Apg2.D(l -{a)y + —

D 48

1/2

(84)

and for a t u r b u l e n t f i lm

V gj

( a ) ( l - ( a » ' D 0.005P£(j£>|(j£)|

D(l -{a)f Apgj

1/2

(85)

Here the negative root is taken when the terna w^ithin the absolute signs beconaes negative. The drift velocity in the form expressed by Eqs. 84 and 85 is convenient for use in analyzing s teady-s ta te adiabatic or thernaal-equilibriuna flows, since in these cases the value of (jr) can be easily obtained

In a general drif t -f lux-model formulation, V • should be expressed in ternas of the naixture velocity Vj- ^ r a the r than (jf), as Vm is the velocity used in the fornaulation. F r o m the definition, we have

( j£)= ( l - ( c v ) ) v ^ - i ^ V na (p ) ^ J " \Mn- i /

(86)

By substituting Eq. 86 into Eq. 84, w e obtain for a laminar film

V 8p-£(a)' J / fiD<Pm>'(l - <' »^

gj= " W ^ l V ^ ' V ' 4.£(.>^Pg ^m + Apg^D 'd - (a))'

48tif

(87)

which is valid for the laminar range given by

(1 - (a))(prn)vm - <Pm^<Jf\r ( » ) p g

^ V, gJ (1 -<^))<Pni>^m + <Pm><Jf>tr

( » > P a (88)

Here the laminar turbulent - t rans i t ion volumetr ic flow is defined by

<JfX = 3200ti£/p£D.

The negative root of Eq. 87 applies when the t e r m within the absolute signs becomes negative. It is easy to show that, for Vgj ^ (l - (o^))(pm^Vj^/ ((a)pg), the flow is cocurrent upward, whereas , for ¥ „ ; l a rge r than the above l imit , the liquid flow is downward.

The solution for the case of turbulent film flow^ is sonaew^hat m o r e complicated. For convenience, let us introduce the following p a r a m e t e r s :

a =

c =

filp g A

0.005(ff)p£(l - (a))^ '

(Q')P

ApgD(l - ( a ) )

0.015p£ •

(89)

-/

Then, for upward liquid flow, we have

r . - r - 2 , , . 2X -.1/2

V

•bvm + [ a v ^ + (a - h')c\

gJ (a - b^)

(v^ 4- c) /2bv.

if a - b^ ^ 0 (90)

na m if a - b^ 0,

which applies under the condition Vm ^ v cb^/a. However, in the t rans i t ion reg ime given by -Vc S. Vj ^ < VcbVa, where the liquid film flow is downward with upward interfacial shear forces on the filna, the vapor-dr i f t velocity becomes

V gJ

b v ^ + [ - a v ^ + (a -F b^)c]

a -I- b^

i /z

(91)

In the range of v given by Vi; ^ ->/c,

•bv m

[av2 - c(a - b2)] 1/2

gJ - w-(92)

which applies to the cocurrent downward flow.

The above solution can be applied only if the following turbulent-flow cr i te r ion is satisfied:

o r

Vgj ^ [(1 - («))<Pm)v^ - (Pm)(j£)^^]/(c>)pg

Vgj ^ [1 - (a))(p„,)vrn + (pm><Jf\^]/<«>Pg_^

(93)

These r e su l t s do not have a very simple form for a turbulent film. However, if the absolute value of the mixture velocity is l a rge , so that the flow is essent ia l ly cocurrent and the gravity effect is smal l , then the turbulent solution can be approximated by the simple form

(1 - (a>)v V na

gJ (c^)pg r5pg[l + 75( l - (a) ) ]

<Pm) I <»)p.

1/2 (94)

Equation 94 for the drift velocity can be t rans formed to obtain the slip ra t io Vg/v£ under the simplifying assumption that the average liquid velocity is much smal le r than the vapor velocity. Then we have

((vg)) ^(V£))

Pf

g

V^) -I 1/2

1 -1-75(1 - (a)) (95)

for an a n n u l a r flow in a p ipe for which 5 = v ( a ) . The above e x p r e s s i o n for s l ip r a t i o i s s i m i l a r to t ha t ob t a ined by Fauske ,^^ n a m e l y , ((vg)) /((v£)) = {Qf/PaV'^, which h a s no d e p e n d e n c e on the void f r ac t ion . The f ac to r t ha t t a k e s the void f r a c t i o n into a c c o u n t in Eq. 94 v a r i e s rough ly frona 0.24 to 1 for the r a n g e 0.8 < Q/ < 1. T h e r e f o r e , for a t u r b u l e n t f i lm , the F a u s k e c o r r e l a t i o n should give r e a s o n a b l y a c c u r a t e r e s u l t s a t h igh void f r a c t i o n s .

The p r e d i c t e d v a p o r - d r i f t v e l o c i t y w a s c o m p a r e d to d a t a f r o m v a r i o u s experiments"*^"^" u n d e r s t e a d y - s t a t e cond i t ions a s shown in F i g . 22. The t h r e e s e t s of da ta c o n s i d e r e d cons t i t u t e about 350 da ta po in t s for a n n u l a r and d r o p - a n n u l a r flow r e g i m e s . T h e o r e t i c a l p r e d i c t i o n s a r e wi th in about ±30% of t h e e x p e r i m e n t a l v a l u e s o v e r a w^ide r a n g e of v a p o r - d r i f t ve loc i t y b e t w e e n 20 and 250 c m / s . A qui te u n i f o r m d i s t r i b u t i o n of the da t a can be a t t r i b u t e d to the u n c e r t a i n t y in the d e t e r m i n a t i o n of void f r a c t i o n in the above e x p e r i m e n t s .

400

200 -

o o

100

80

I 60 £ Q.

40

—

—

-

-

-

—

/

/

1 1 1 T

-30% nA X / o i S

AT/ / y + 30 %

/ \ I I I

1 1

x ^ ^

^x^y

V

1 1

1

D

i '^

\ .

' / /

/ / /

^

Sa

D D

D 0

• A

o

w°° D

GILL-LAMINAR GILL-TURBULENT ALIA-LAMINAR ALIA-TURBULENT CRAVAROLO-LAMINAR CRAVAROLO -TURBULENT

1 1

>^

/

-

-

1 . 20 40 60 80 100 200

CALCULATED DRIFT VELOCITY, cm/s 4 0 0

Fig. 22. Comparison of Annular-flow Correlation with Experimental Data. ANL Neg. No. 900-5768.

We note he re that these data include a considerable number of points taken in the drop-annular flow regime at modera te r a t e s of gas flow. This indicates that the p resen t vapor-dr i f t -veloci ty cor re la t ion can be used both for ideal annular flows w^ithout entra inment as w^ell as for annular mis t flows with modera te to low entrainment . However, as the amount of liquid entrained in the gas core becomes la rge at high r a t e s of gas flow, ' ^ the measu red vapor-dr i f t velocity s ta r t s to depart considerably from the predic ted values and the p resen t analysis overpred ic t s these values.

The drif t-velocity corre la t ion for the annular flow has been expressed in t e r m s of the mixture velocity, since v^^ is the basic var iable in the formulation of the general drift-flux model. However, it is also in teres t ing and important to reso lve the express ion for Vgj in ternas_of the total volunaetric flux ( j ) , since (j) was the var iable used to co r re l a t e Vgj in d i spersed two-phase flow reg imes .

By considering the turbulent film-flow reg ime and using the definition (jf) = (l - (of))(j) - (ci?)Vgj, we can resolve Eq. 85 for the mean drift ve locity Vgj. The resu l t does not have a simple form; however, for most p rac t i ca l cases , it can be approximated by a l inear function of ( j ) :

1 - { < y ) V g j

(a) + 1 -F 75(1 - (a)) pg"

! Jl^ Pf.

1/2 ( j ) + Apg^D(l - (a))

0.015p£ (96)

This express ion may be further simplified for p„/p£ « 1 as

/ 1 - (a) ^ ~ Y. gJ

(a) + 4vfg/p£J'-(i) +

Apg„D( l - iot))

0.015p£ (97)

F r o m the conaparison of Eq. 97 to Eq. 58, the apparent distr ibution p a r a m e t e r for annular flow becomes

Co - 1 + 1 - ( a )

.; ( P g / p £ « 1). (a) + 4 . ^ g / p £

This indicates that the apparent Cg in annular flow^ should be close to unity.

C. Annular Mist Flow

(98)

As the gas velocity i nc reases in the annular flow , the entra inment of liquid from the film to the g a s - c o r e flow takes place. Based on recent ly de veloped c r i t e r i a for an onset of entrainment, '^ the cr i t ica l gas velocity for a rough turbulent film flow can be given by

f o r N^f ^ j 5

(99)

0.1146 for Np,f > 15'

w h e r e NM£ = t i£/[p£ava/gAp] . H o w e v e r , in g e n e r a l , the v a p o r flux i s nauch l a r g e r than the l iquid flux in the a n n u l a r - m i s t - f l o w r e g i m e ; then , for a weak ly v i s c o u s fluid such a s w^ater o r s o d i u m , the above c o r r e l a t i o n naay be r e p l a c e d by

ijgi - bi > (^T^r- (100)

If Inequa l i ty 100 i s s a t i s f i e d , t hen the d r o p l e t e n t r a i n m e n t in to the g a s - c o r e flow should be c o n s i d e r e d ; o t h e r w i s e the c o r r e l a t i o n for a n n u l a r flow^, Eq. 97 , can be appl ied .

The c o r r e l a t i o n for Vgj in a n n u l a r m i s t f lows can be r e a d i l y deve loped by combin ing the p r e v i o u s r e s u l t s for a d i s p e r s e d flow and p u r e a n n u l a r flow. The a r e a f r a c t i o n of l iquid e n t r a i n e d in the g a s c o r e f r o m to t a l l iqu id a r e a at any c r o s s s e c t i o n i s deno ted by Ed, and the c r o s s - s e c t i o n a l - a r e a - a v e r a g e d void f r a c t i o n by (ex). Then the f i l m - a r e a f r a c t i o n i s g iven by

l i q u i d - f i l m c r o s s - s e c t i o n a l a r e a

*^°^^ t o t a l c r o s s - s e c t i o n a l a r e a

and the m e a n l i q u i d - d r o p l e t f r a c t i o n in t h e gas c o r e a lone i s g iven by

c r o s s - s e c t i o n a l a r e a of d r o p s (1 - '° ' ))Ejj °^drr,r, = Z- i r = 7 ^ • (102)

'-'^"P c r o s s - s e c t i o n a l a r e a of c o r e i _ (^ _ ( a ) ) ( i - E H )

C o n s e q u e n t l y , Q'core should be u s e d in the a n n u l a r - f l o w c o r r e l a t i o n , Eq. 97 , to ob t a in the r e l a t i v e m o t i o n betw^een the c o r e and the f i lm , w h e r e a s Q?drop should be u s e d in the d ispersed- f low^ c o r r e l a t i o n to ob ta in a s l ip b e t w e e n d r o p l e t s and g a s - c o r e flow.

By denot ing the g a s - c o r e ve l oc i t y , l i q u i d - d r o p v e l o c i t y , and filna v e l o c i t y by Vg^, vf^,, and V£f, r e s p e c t i v e l y , t h e t o t a l v o l u n a e t r i c flux i s g iven by

^j> = [vgc( l - o^drop) + ^dropVfclo 'core + Vff(l - Q^core)- ( l 03 )

F u r t h e r m o r e , by deno t ing the t o t a l v o l u n a e t r i c flux in the c o r e b a s e d on the c o r e a r e a by j r n r e ' ^^ h a v e f r o m t h e a n n u l a r c o r r e l a t i o n , Eq. 97 ,

44

J core (j)^ (1 -<°»( i - fd )U , , A ^ ^ " " / J f ' - ^^>;. (104)

( a ) + 4 .7^g/p£ 0.015p£

On the other hand, from the dispersed-f low cor re la t ions , it can be shown that, for a d is tor ted-drople t or churn-drople t flow reg ime , the drift velocity can be given approximately by

/agAp ^^Vg)) - j ^ ^ . ^ ^ = 7 2 f - ^ -

1 / 4 Ed(l -<c^))

'core Pg / (a) -F E d ( l - (a)) (105)

Here w e have used an approxinaation based on (l - (ex)) « 1. How^ever, depending on the co re -gas velocity, the dispersed-flow^ drif t-velocity c o r r e lation for a much smal ler par t ic le should be used. When the droplets a re generated by the entrainment of liquid film, the following approxinaate form is suggested for an undis tor ted-par t ic le reg ime outside the Stokes regime:'*^

( ( v „ ) ) - j ' g core

0.5r, (_gAp)'

^^gPg

n i / 3 Ed(l - M )

(a) + Ed(l - (a)) ' (106)

where the par t ic le rad ius naay be approxinaated from the Weber-number c r i ter ion at the shearing-off of wave c r e s t s . Thus,

6CT J _

P g < j > (107)

The above re la t ions apply only when the total volumetr ic flux is sufficiently high to induce fragmentations of the wave c r e s t s . Hence, Eq. 106 should be used when

l(j)l > 1.456, /agAp

'g

,1 /4

rCrvu/gAf

• 1 / 1 2

(108)

By conabining the above r e su l t s , we obtain

(1 - ( a ) ) ( l - Ed)

SJ ( a ) + 4 ^ 0 / p f ( i ) +

Apg^Dd - ( a ) ) ( l - Ed)

0.015p£

<

JlfAhl .1 /4

+ E d ( l - < ^ » ^ ,

^ (a) -f- E d ( l - (a)) " " ' or

3a

Pg

"(gAp)'

^ g ° g j

fH/3

(\f

(109)

where the la t ter express ion applies under the condition given by Eq. 108. If the radius of the par t ic le is very small , then the essen t ia l contribution to the re la t ive naotion betw^een phases comes from the f i rs t t e r m of Eq. 109, and the core flow may be considered as a homogeneous d ispersed flow. In such a case, Eq. 109 reduces to

(1 - <a>)(l - E ^ ) / /Apg^Dd - <a»(l - E<jA

This express ion shows a l inear dec rease of drift velocity in t e r m s of ent ra ined liquid fraction, which can be observed in var ious exper imental data.^^'^°

VII. COVARIANCE OF CONVECTIVE FLUX

In the one-dimensional drift-flux model, the naonaentum and energy convective fluxes have been divided into th ree t e r m s : the mixture convective flux, the drift convective flux, and the covariance te rm, as can be seen from Eqs. 54 and 55. In other words , the convective flux of quantity i for the mixture can be wri t ten as

3 / V / , \ \ S / T - ^ 3 /^Q'd^PdPc - \ 3^(^l<°'kPk*kVk>j = 9^(Pm*mVm) + 5^1^ <p^) ^'^dcVdjj

+ fz Z^O^(°'kPk*k^k)' (111) k

where A lfdc ~ ^^*d^^ - «*c^^ and COV(ai^Pk'lfkVk) = ^<^kPk^k(vk -^(v^)))) . Therefore , for the momentum flux, we have ^\^ = v^ and Aijfdc ~ Vdi / ( l - (o'd)). On the other hand, for the enthalpy flux, we have \|i] = hj^ and A jidc = ((hd)) - ((h^,)), which is equivalent to the latent heat if phases a re in t h e r m a l equilibrium.

To close the set of the governing equations, we must specify re la t ions for these covariance t e r m s . This can be done by introducing distr ibution paranae ters for the naonaentuna and energy fluxes. If we define a distr ibution p a r a m e t e r for a flux as

_ <°^k'^k^k> , .

""^^ = (ak)((*k)>«vk)>' ^^^^^

the covariance t e r m becomes

COV(pkQ'k'^kVk) = Pk^Mk(vk - « v k » ) ) = (C^k - l)Pk<C'k>«*k»«Vk»- (113)

For the momentum flux, the distr ibution p a r a m e t e r is defined by

(ak)((vk))

Physical ly, Cvk r e p r e s e n t s the effect of the void and naonaentuna-flux profi les on the c r o s s - s e c t i o n a l - a r e a - a v e r a g e d momentuna flux of k phase. A quantitative study of Cvk ^an be made by considering a symmet r i c flow in a c i rcu lar duct and introducing the power- law express ions in pa ra l l e l with the analysis of Cg in Sec. VI. A. Hence w e postulate that

Q'k - Q^kw

«ko - °'kw 1 - ( R / R j " (115)

and

- ^ = 1 - ( R / R J " ^ , (116) Vko

where the subscr ip ts o and w refer to the value at the center l ine and at the wall of a tube.

For simplicity, it is assunaed that the void and velocity profi les a re s imi la r ; i .e . , n = na. This assumption is widely used in naass - t ransfer p rob lems , and it may not be unreasonable for fully developed two-phase flow^s if one considers that the vapor flux and, hence, the void concentration great ly influence the velocity dis tr ibut ions. Under this assumption, it can be shown that

n + 2 / , 3 n \ / , n \

Cvk = . (117)

(^kw + ^ ° ' k ^ i )

where

^°'k = o-ko - ° 'kw

For a d i spersed vapor phase , o gw * * ^^'g' hence.

On the o t h e r hand , f rom Eq. 62 , the v o l u m e t r i c - f l u x - d i s t r i b u t i o n p a r a m e t e r CQ b e c o m e s

Co n + 2 n + 1

(119)

T h e r e f o r e , in the s t a n d a r d r a n g e of n, the p a r a m e t e r Cvg can be g iven app r o x i m a t e l y by

^vg 1 + 0.5(Co - 1). (120)

F o r a l iquid p h a s e in a v a p o r - d i s p e r s e d - f l o w r e g i m e , ^f-w " 1 ^-^^ Qfg < 1. T h e n f r o m Eq. 117 i t can be sho-wn tha t , for a s t a n d a r d r a n g e of O!^^ in the bubb ly - and c h u r n - f l o w r e g i m e s , Cvf can be a p p r o x i m a t e d by

Cvf = 1 + 1.5(Co - 1). (121)

F o r an a n n u l a r flow, the m o m e n t u m c o v a r i a n c e t e r m can a l s o be c a l cu la ted by u s ing the s t a n d a r d ve loc i ty p r o f i l e s for the v a p o r and l iquid f lows. Thus w e ob ta in

•'vk

'1.02 ( t u r b u l e n t flow)

1.33 ( l a m i n a r flow). (122)

The above r e s u l t for the ind iv idua l p h a s e s can now be u s e d to s tudy the m i x t u r e c o v a r i a n c e t e r m . By def ining the m i x t u r e - m o m e n t u m - d i s t r i b u t i o n p a r a m e t e r as

' v m CvdPd<Q^d^ + CvcPc^Q'c^

<Pm> (123)

t h e c o v a r i a n c e t e r m b e c o m e s

ZcOV(akPkv|,) = (C k

v m 1) v ^ PcPd<°'d> -

V (1 -<«d»<Pm> ^^

^PcPd<°^d>

<Pm> (C vd V c ) v m V d j (124)

In v i ew of the above a n a l y s i s , t h e o r d e r of m a g n i t u d e of (C^d " ^-vc) i s the s a m e a s t ha t of (C^j^^ - l) o r l e s s ; t h e r e f o r e , t h e l a s t t e r m on the r i gh t -hand s ide of Eq. 124 can be n e g l e c t e d for a l m o s t a l l c a s e s . T h i s t e r m m a y be i m p o r t a n t only in the n e a r c r i t i c a l r e g i m e and if Vj^ = ^ d j - H o w e v e r ,

in general , Vji becomes insignificant as the density ra t io approaches unity; hence, under the above conditions the convective t e r m itself becomes relat ively small . Consequently, even for this case , the t e r m may be dropped. Thus we have

^ C O V ( a k P k v y - (Cvm - 1)

k

PcPd^^'d^ - 2 (p >v2 + ^ " ^ V^-(1 - SQ'd'')vPm''

(125)

The va lue of C^j.^ can be e v a l u a t e d f r o m Eq. 123 by u s ing E q s . 120 and 121 o r Eq. 122. In the bubb ly - and c h u r n - f l o w r eg innes of p r a c t i c a l i m p o r t a n c e , C v m can be g iven a p p r o x i m a t e l y by

^vm 1 + 1.5(Co - 1). (126)

H o w e v e r , in t h e n e a r c r i t i c a l r e g i m e C v m d e p e n d s a l s o on the void f r a c t i o n and the d e n s i t y r a t i o . F u r t h e r m o r e , at v e r y low void f r a c t i o n s in a fully d e ve loped flow o r in a deve lop ing flow, the v a l u e of C v m should be r e d u c e d to the one for the s i n g l e - p h a s e flow given by Eq. 122. The effect of the d e v e l o p m e n t of the vo id p ro f i l e in to t h a t g iven by the pow^er lav/ m a y be t a k e n in to accoun t by a s i m i l a r v o i d - f r a c t i o n c o r r e c t i o n t e r m u s e d in the c o r r e l a t i o n for Co in Eq. 70. By r e c a l l i n g t h a t for a t u r b u l e n t flow C v m =" 1-0 at a -- 0, w e ob ta in for a round tube

Cvm - 1 + 0.3(1 - y p g / p f ) ( l - e-i«<^>), (127)

•which m a y be u s e d both for a fully deve loped flov/ and for a deve lop ing flow^.

On the o the r hand , for a t u r b u l e n t - a n n u l a r - f l o w r e g i m e , we h a v e , irom E q s . 122 and 123, C.^^^ = 1.02. F o r a l l p r a c t i c a l p u r p o s e s , t h i s m a y be f u r t h e r a p p r o x i m a t e d by

•-vm 1. (128)

In r e a l i t y , the t r a n s i t i o n f r o m the v a l u e given by Eq. 127 to t ha t g iven by Eq. 128 i s a g r a d u a l one t h r o u g h the c h u r n - a n n u l a r (or s lug-annular ) - f low^ r e g i m e in wh ich c h a r a c t e r i s t i c s of c h u r n and a n n u l a r f lows a l t e r n a t e . If a s ing le c o r r e l a t i o n for Cvm i s p r e f e r r e d , r e g a r d l e s s of t h e f l o w - r e g i m e t r a n s i t i o n s , t h e n Eq. 127 m a y be safe ly e x t r a p o l a t e d into h i g h e r - v o i d - f r a c t i o n r e g i m e by a s i m p l e mod i f i ca t i on given by

Cvm = 1 + 0.3(1 - V p 7 7 f ) [ l - e-i«<o'Xi-<°'»]. (129)

A s i m i l a r a n a l y s i s can be c a r r i e d out for the e n t h a l p y - c o v a r i a n c e t e r m by a s s u m i n g the void , v e l o c i t y , and en tha lpy p r o f i l e s . In g e n e r a l .

X COV(ai,PkhkVk) = (Chm - iXPm^^m^m k

" f " ^ ? [-(Che - l ) « h c » + (Chd - l )«hd»]Vdj . (130)

w^here

fChm = Z^hkPk<<^k>«hk»/«Pm>hm) k

Chk = (akhkVk>/(<o'k>«hk»<<vk»).

(131)

For a thermal-equil ibr iunn flow, h„ = hgg and hf = hfs ' where h„s and hfg a re the saturat ion enthalpies of vapor and liquid. Since in this case the enthalpy profile is completely flat for each phase, the distr ibution par a m e t e r s become unity; i .e . , Chg = C^f = C^n-i = 1. It is also evident that if one of the phases is in the saturated condition, then Cjik f°^ that phase becomes unity.

On the other hand, in the s ingle-phase region, the distr ibution par a m e t e r can be calculated from the assumed profiles for the velocity and enthalpy. Using the s tandard power- law profiles for a turbulent flow, i .e. , v/vo = (y/R)^/" and (h - hw)/(ho - h^) - (y/R)^/"^, where y is the distance from the w^all, we can show^ that the covariance t e r m is negligibly smal l both for developing and fully developed flows.

F r o m the above two limiting cases , we can conclude that the enthalpy covariant t e r m may become important only in highly nonequilibrium flow. Even in that case , the energy associa ted w^ith phase change is considerably l a rge r than that associa ted with changes in t r a n s v e r s e t e m p e r a t u r e prof i les ; there fore , except for highly t rans ient cases , the enthalpy covariance can be neglected. Hence,

^ y COV(cVkPkhkVk) = 0. (132) ^^ k

VIII. FLOW-REGIME TRANSITION AND DRIFT VELOCITY IN VERTICAL SYSTEM

In the preceding sect ions, the drif t-velocity cor re la t ion has been de veloped for var ious two-phase flow reg imes . To use this cor re la t ion , t h e r e fore, w e miust identify the flow reg ime f i rs t and then choose a suitable express ion for Vdj- In the l i t e ra tu re for two-phase flow^ severa l c r i t e r i a for

f low-regime t rans i t ions a re summar ized in Refs. 46 and 55-58. These existing c r i t e r i a may be used to identify the flo>v r eg imes under var ious conditions. How^ever, in Avhat follows, w e shall develop flow^-regime t rans i t ion c r i t e r i a that a re based on the re la t ive motion bet^veen phases and a r e consistent w^ith the concept of the drift-flux naodel.

In forced-convection boiling sys tems , four flow reg imes a re of p rac t ica l importance:^^ churn-turbulent bubbly, annular, annula r -mis t , and droplet or l iqu id-d ispersed reg imes . In the vapor-d ispersed- f low reg imes such as the bubble, d is tor ted bubbly, and churn-turbulent flows in forced-convection s y s t e m s , the effect of the concentration and velocity profiles dominates the re la t ive motion between phases . This can be seen from the importance of the t e r m associa ted with the distr ibution p a r a m e t e r with r e spec t to the local drift effect in Eq. 58. Consequently, var ious vapor -d i spersed- f low reg imes can be sat isfactori ly represen ted by the churn-turbulent-f low corre la t ion alone. This is also shown by the exper imenta l data in Figs . 6, 7, 18, 19, and 20.

The t rans i t ion c r i t e r ion between churn-turbulent bubbly flow and annular flow^ is obtained by postulating tw o different mechanisms for the t rans i t ion:

1. Flow r e v e r s a l in the film section along large bubbles and following t r anspor t of liquid m a s s in the slug or wake regions into the film section.

2. Destruction of liquid slugs or la rge waves by entra inment or de formation and subsequent redis t r ibut ion of liquid m a s s into the droplets and film.

The f i rs t mechanism as sumes that, for the film section along la rge bubbles, the annular drif t-velocity corre la t ion can be used locally. Hence, by setting the f low-reversa l condition as (jf) = 0 and using Eqs. 46 and 48, we obtain, from Eq. 96,

{a) V^' <ar "-^ —I = - ( a > - 0 . 1 , (133)

t0.015[l + 75(1 - (a))]j

where j * = jg/v ApgD/Pg. Here the above approximation is a good fit to the exact form for (o/) < 0.93. At this point, the flow is st i l l in the churn-turbulent regime. It has to support ent ire liquid fraction in the film in o rder to have a t ransi t ion; therefore the mean void fraction is calculated from the drif t-velocity correla t ion for the churn-turbulent flow. Thus fronn Eq. 76, (a) can be obtained in t e r m s of ^jg) and (jf). If this express ion is substituted into the above equation, we get

f= ^ • - 0 . 1 . (134)

In g e n e r a l , the l iquid flux and the t e r m i n a l ve loc i t y a r e nnuch s m a l l e r than the g a s flux; t hus Eq. 134 can be f u r t h e r s imp l i f i ed to

it = TT - 0 - 1 . (135)

w h e r e Cg i s g iven by Eq. 68 o r 71 . T h e r e f o r e , for a round t u b e .

Ja - L. / j ^ - ^ . - 0 . 1 . (136) g ^ g V AogD , , ^ ^ r—; 1.2 - 0 .2VPg/ g/Pf

The r a n g e of j j i s a c c o r d i n g l y f r o m 0.73 to 0.9. The above va lue i s c o n s i s t e n t wi th v a r i o u s e x p e r i m e n t a l o b s e r v a t i o n s . ^ ^ " ^ F o r e x a m p l e , Wallis^^ found jt to be 0 . 8 - 0 . 9 , and B e n n e t t et al . o b s e r v e d the i n c r e a s e in j„ a t h i g h e r p r e s s u r e s t ha t i s p r e d i c t e d by Eq. 136.

The s e c o n d m e c h a n i s m of the c h u r n - a n n u l a r f low- reg inne t r a n s i t i o n , v/hich ^vas f i r s t s u g g e s t e d by Dukle r and Smi th , s e e m s to be m o r e a p p l i c a b l e to a flow^ in a l a r g e r - d i a m e t e r tube . In t h i s c a s e , the f o r c e b a l a n c e on the wave c r e s t r a t h e r t han the l iquid naotion in the f i lm d e t e r m i n e s the t r a n s i t i o n . S ince the i n t e r f a c i a l g e o m e t r i e s a r e e x t r e m e l y r o u g h a t the c h u r n - a n n u l a r f low^-regime t r a n s i t i o n , the wâve d e f o r m a t i o n o r e n t r a i n m e n t c r i t e r i o n for a r ough t u r b u l e n t r e g i m e deve loped by I sh i i and Gro lmes^^ m a y be used . In t h i s c a s e , the c r i t e r i o n i s g iven by Eq. 99- Howêver, for a n o n v i s c o u s fluid such a s wâter and s o d i u m , i . e . , Nn,f < l / l 5 , Eq. 99 can be r e w r i t t e n a s

w^here the n o n d i m e n s i o n a l gas flux K i s the K u t a t e l a d z e n u m b e r and N^r = lJ,£/[p£aâ/gAp]^/^. The v a l u e of K does not v a r y m u c h for nnost n o n v i s c o u s f lu ids , be ing about 3. The second c r i t e r i o n i s va l id if t he p r e d i c t e d gas flux b a s e d on the K u t a t e l a d z e n u m b e r i s s m a l l e r t h a n the one b a s e d on t h e f i r s t c r i t e r i o n . T h i s o c c u r s if the followîng condi t ion i s s a t i s f i e d :

D > . / T T - N . f - / ^ . (138)

F o r wâter at low p r e s s u r e , t h i s c o r r e s p o n d s a p p r o x i m a t e l y to D > 6 cm.

The p r e s e n t m o d e l g ives two d i f fe ren t c r i t e r i a for the c h u r n - a n n u l a r f l o w - r e g i m e t r a n s i t i o n : one depend ing on t h e d i a m e t e r of t h e t u b e and the o the r depend ing only on t h e p r o p e r t i e s . T h i s d iv i s i on i s q u a l i t a t i v e l y c o n s i s t e n t

wîth the observat ion of Pushkina and Sorokin •* on the breakdow^n of liquid film, and of Wallis and Kuo on the liquid downward penetrat ion.

The t rans i t ion betwêen the annular and annu la r -mis t flow can be given by the onset of entrainment.^^ Hence, for nonviscous fluid.

ljgl> ' ^gApV^, -o .E

N Pf

-1/3 11.78Re£ ; Ref ^ 1635,

1; R e f > 1635, (139)

where Re£ = p£|(J£)1D/|JI£. Figure 23 compares the above c r i te r ion to the experinnental data.

In an adiabatic sys tem, the t rans i t ion from an annular flow to a droplet flow is gradual. As the fraction of liquid entrained, E^j, i n c r e a s e s , the cha r ac te r i s t i c of the flow changes from that of a separa ted flow to a d i spersed flow. This t rend is c lear ly exhibited by the express ion for the drift velocity in the annular-mist-flow^ reg ime, Eq. 109. In other wôrds, the w^hole annular-mis t flow can be considered as a t rans i t ion reg ime. On the other hand, in the diabatic sys tem, the occur rence of the cr i t ica l heat flux and subsequent drying out of the wâll introduces a sudden change in flow^ reg imes . If this occurs , Eq. 109 should be used with 100% entra inment or E^j = 1.

The recommended drift-velocity corre la t ions for a ver t ica l -boi l ing flow^ system a re given in Table II.

100 1 — I I I I 1 1 1 1 1 I I I M I 1 I~l I M I I

INCEPTION CRITERIA GIVEN

~l I I M I U

_1_- I - I I I i I I

LIQUID REYNOLDS NUMBER, Re =

10"

Fig. 23. Comparison of Data for Onset of Entrainment according to Inception Criteria. ANL Neg. No. 900-4572 Rev. 1.

Flow Regime

Churn-turbulent

Annular

Annular-mist

Liquid dispersed

TABLE II Drift Velocity for

Transit ion Cr i ter ia (Applicable range)

M'V^°fe-»') and

|'.|-(':r) "••'"

J^{i,-')<w<{^)"\r

l%l'(°1°)"V,' "•' •:-•'

Ed = I 0 or q;; > q^,.

Vertical Boiling Sy stem

Vapor-dnf t Velocity, Vgj

where

Co = { I 2 0 - O 2 y p g / P f ) l l - e - ' »^^ ' )

and

Ny.£ = Hf/(P(aVo/gAp) 1/2

r-

I -(a)

^' (a) + 4VPg/Pf ^^. /

_, ApgDd - (c())

V O O l S P f

r-

(1 -<cr»(l - Ed)

^ (Q' + 4ypg/pf <j>+ ^

Edd - ("»

<c«) + Edd - <o>)

/ApgDd - <a))(l

0 015pf

' ^ / c ^ g A p y ' ^ V Pg /

3a

/ g

(gAp)*

L^g'^gj

1/3 1

The second expression applies when

1 J | > 1 456 - S j - -V p | / VP

Vgj - d - (a-)) X.

-s ' goVa/gApy

- l / U

r^/agApy/^ V p | /

or 3o

[Pg (gAp)'

^'gPg

1

J <J>^

- E d )

IX. CONCLUSIONS

For a d i spersed tw^o-phase flow sys tem, the re la t ive motion between phases has been analyzed by considering the drag force, the gravitat ional force, and the effect of the p r e s s u r e gradient due to shear s t r e s s e s , and using a s imi la r i ty hypothesis based on the Reynolds number and drag coefficient. The effect of the mul t ipar t ic le sys tem is taken into account by using the mixture viscosi ty in the Reynolds number. The p resen t model for the re la t ive motion between phases has been developed for genera l d i spersed two-phase flows; therefore i ts applicability is not l imited to par t icula te flow^s, but it can also be applied to bubbly and droplet flow^s. For so l id-par t ic le sys t ems , it ag rees with experinnental data at all Reynolds-number and par t ic le -concent ra t ion ranges of p rac t ica l in teres t . Fur thernnore , the p resen t theory can be reduced to the existing theore t ica l model for a so l id-par t ic le systenn in the Stokes reg ime by considering the l imit ing case of smal l pa r t i c l e s .

On the other hand, for f luid-part ic le sys tems , considering bubbly flows in a ve r t i ca l channel, the p resen t model can be essent ia l ly reduced to conventional semiempi r i ca l corre la t ions . Consequently, the re la t ive motion betwêen phases in d i spersed twô-phase flow^s can be predicted by the unified theory for both solid- and f luid-part ic le sys tems at all ranges of Reynolds numbers .

For annular flow, the constitutive equation for the drift velocity vras developed by taking into account the effect of gravity, interfacial shear s t r e s s with i ts dependence on interfacial roughness , and flow reg imes in the liquid film. The constitutive equation obtained in this r epor t is cast into two different fo rms : one useful for analyzing s teady-s ta te flows and the other for t rans ien t flow^s. The predicted vapor-dr i f t velocity from this study was compared to about 350 data points from var ious exper iments . Although the data used in the comparison included those taken in the drop-annular-f low reg imes with modera te entrainment , the theore t ica l predict ions were within ±30% of the nneasured values. However, -when the amount of liquid entrainnnent wâs la rge , the present corre la t ion overpredic ted the vapor-dr i f t velocity. However , this t rend can easily be explained by a new^ly developed corre la t ion for an annular dispersed-f low regime that shows an almost l inear dec rease of drift velocity in t e r m s of entrained liquid m a s s .

The presen t constitutive equations for the drift velocity in various two-phase-flow reginnes have been obtained from the s teady-s ta te and adiabatic fornnulations. The effects of heat t ransfer and phase changes on the drift ve locity wêre considered secondary. These effects appear only indirect ly through the local va r iab les , such as the void fraction and the mixture velocity in the drift constitutive equation. It is a common prac t ice to apply constitutive r e lat ionships obtained under s teady-s ta te conditions to the t rans ien t p rob lems , wîth an assunnption that the paranneters entering into a constitutive re la t ionship a re local var iables and are functions of t ime. Therefore , the application of the p resen t constitutive equation for the drift velocity to the t r ans ien t two-phase flow with a phase change will be consistent with the common pract ice .

However, the basic assumption of the drift-flux model i s that a s trong coupling exists between the motions of two phases . Therefore , cer ta in two-phase problems involving a sudden accelera t ion of one phase nnay not be appropriately descr ibed by this model. In these cases , iner t ia t e r m s of each phase should be considered separa te ly , that i s , by use of a two-fluid model. However, the r ea l usefulness of the drift-flux nnodel in many prac t ica l engineering problems comes from the fact that even two-phase mix tures that a re wêakly coupled locally a re strongly coupled when considered as a total system. This is because the relat ively large axial extent of the sys tems usually gives sufficient interact ion t imes for the momentunn exchange between two phases .

A P P E N D I X

R e l a t i v e Mot ion in S i n g l e - p a r t i c l e S y s t e m

A m o t i o n of t h e s ing l e so l id p a r t i c l e s , d r o p s , o r bubb l e s in an inf ini te m e d i u m h a s b e e n s t u d i e d e x t e n s i v e l y in t h e p a s t . (See, for e x a m p l e , Re f s . 52-54 . ) In wha t fo l lows , we s h a l l s u m n n a r i z e t h e s e r e s u l t s in s i m p l e f o r m s usefu l for t h e d e v e l o p m e n t of the dr i f t c o n s t i t u t i v e equa t ion in m u l t i p a r t i c l e s y s t e m s .

By denot ing t h e r e l a t i v e v e l o c i t y of a s i ng l e p a r t i c l e in an inf ini te m e d i u m by Vrm = v^j - v^^ , we define t h e d r a g coeff ic ient by CDOO = -2Fj-) / P " rco l^rool'^^d' w h e r e T-^ i s t he d r a g f o r c e and r i s t he r a d i u s of a p a r t i c l e . On the o the r hand , t h e s u m of the p r e s s u r e and body f o r c e s ac t ing on the p a r t i c l e is g iven by

F + F p g l^^d(Pc - Pd)g (A . l )

wh ich shou ld be b a l a n c e d by t h e d r a g f o r c e . H e n c e , F p + F g + F j ) = 0. By i n t r o d u c i n g the n o n d i m e n s i o n a l p a r a m e t e r s for t h e v e l o c i t y f i e lds and r a d i u s g iven by V* s | v | (p^/iJ,cgAp)^^^ and r ^ = r^(p^gAp/|a^) b a l a n c e for the r a d i u s as

1/3 we can so lve t h e f o r c e

r * = lCr>^v*^ roo-

(A.2)

T h e s t a n d a r d p a r t i c l e R e y n o l d s n u m b e r and the v i s c o s i t y n u m b e r a r e def ined by

a n d

N ^^dPci

Reo°

N,, =

= 2 r J V * d roo

/ / y — \ ^

(A. 3)

w h e r e Nn m e a s u r e s t h e v i s c o u s f o r c e i n d u c e d b y a f l o w t o t h e s u r f a c e - t e n s i o n 66 f o r c e . E x t e n s i v e s t u d i e s of t h e s i n g l e - p a r t i c l e d r a g s h o w t h a t , i n g e n e r a l ,

t h e d r a g c o e f f i c i e n t i s a f u n c t i o n of t h e R e y n o l d s n u m b e r . H o w e v e r , t h e e x a c t f u n c t i o n a l f o r m d e p e n d s o n w h e t h e r t h e p a r t i c l e i s a s o l i d p a r t i c l e , d r o p , o r b u b b l e .

F o r a v i s c o u s r e g i m e , t h e f u n c t i o n Cj-jgo i s g i v e n b y

2 4 'Doo N R e »

( l + 0 . 1 N « j , ^ ) . (A. 4)

For solid pa r t i c l e s , the drag coefficientbecom.es essent ia l ly constant at approximately Cjvo = 0.45 for Nj^ ^ 1000. This Newton's r eg ime holds up to NReoo = 2 X 10^

For fluid par t ic les such as drops or bubbles, the flow reg ime is charac te r i zed by the distort ion of par t ic le shapes and i r r egu l a r motions. In this d i s tor ted-par t ic le reg ime, the exper imenta l data show that the t e rmina l velocity is independent of par t ic le size. From, this it can be seen that the drag coefficient C j ^ does not depend on the viscosi ty , but should be proport ional to the radius of the par t ic le . Physically this indicates that the drag force is governed by the distort ion and swerving motion of the par t i c le , and the chajige of the par t ic le shape is toward an inc rease in the effective c r o s s section. Therefore , Cjy^ should be scaled by the mean radius of the par t ic le r a the r than the Reynolds number. ^ Then,

^Doo ir^VjZ;/^ (or C^ = fNj^^*) (A. 5)

for Nn ^ 3 6 A / 2 ( 1 + 0. lNj^^^)/N|^g<„. However, since the t e r m i n a l velocity in this regime can be uniquely re la ted to physical p rope r t i e s , Eq. A. 5 can be rewri t ten in t e r m s of the t e rmina l velocity or the Reynolds number as

Ci>o = ^ N ^ - | v * „ ( - ^I^ - ^ ^ ^ R e - ) - (^-^^

Note that the above express ion is a special form based on the solution of the balance between drag and gravity forces and it may not be used in m o r e general situations. As the size of bubbles further i nc r ea se s , the bubbles be come spher ica l -cap shaped and the drag coefficient reaches a constant value of C-Q^ = 8/3 . The t rans i t ion from the dis tor ted-bubble reg ime to the spher ica l -cap bubble reg ime occurs at around r^ = 2/NM'' ' . Fo r a liquid drop, the drag coefficient can inc rease further according to Eq. A. 5; however, eventually a droplet becomes unstable and dis integrates into smal le r drops. This l imit can be given by the well-known Weber-number cr i te r ion , and it cor responds to r^ s 3/NM''^ and Cjyxi ^ 4. The above resu l t s on the drag coefficient for single par t ic les of solid, liquid, and gas a re summar ized in Fig. 1.

By knowing the drag law, Cj^^, = ^Dool-'^Reco)' '^^ ^^^ calculate the t e rmina l velocity from Eq. A.2). In the viscous reg ime, the t e rmina l velocity can be approximated by

roo ^ [ ( 1 + 0.08r*V^^ - 1]. (A.7)

On the other hand, in Newton's regime for solid pa r t i c l e s , the drag coefficient is constant; therefore .

http://coefficientbecom.es

a/2 V* = 2 . 4 3 r * ^ ' ^ (A.8) poo

which holds for r^j s 34.65.

For the d is tor ted-f lu id-par t ic le reg ime, the t e rmina l velocity reduces to a constant value of

v*„ = -/Z/^\1\ (A. 9)

Hence, in this reg ime, the relat ive velocity is independent of the fluid-par t ic le size. F u r t h e r m o r e , for the spher i ca l -cap bubble reg ime, the t e rmina l velocity becomes

vL = ^f" (A. 10) roo

These resu l t s a re summar ized in Fig. 2.

58

ACKNOWLEDGMENTS

I would l ike to e x p r e s s m y a p p r e c i a t i o n to Dr . N. Z u b e r of NRC for v a l u a b l e d i s c u s s i o n s on t h e sub jec t . I a m a l s o i n d e b t e d to D r s . M. A. G r o l m e s , D. Condiff, and T. C. C h a w l a for hav ing d i s c u s s e d v a r i o u s a s p e c t s of t h e a n a l y s i s .

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water reactor safety- research--analysis development (nrg

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